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mpute the mean and standard deviation of the uniform distribution on [a, b]. (See exam- ple 9.8.3.) ⇒ 4. What is the expected value of one roll of a fair six-sided die? ⇒ 5. What is the expected sum of one roll of three fair six-sided dice? ⇒ 6. Let µ and σ be real numbers with σ > 0. Show that N (x) = 1√ − (x−µ)2 2σ2 ... |
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... ............. ............ ............ • ............. ............. ............. ............. Figure 10.1.2 A cardioid: y = 1 + cos x on the left, r = 1 + cos θ on the right. Each point in the plane is associated with exactly one pair of numbers in the rectangular coordinate system; each point is associated wit... |
EXAMPLE 10.2.2 We find the second derivative for the cardioid r = 1 + cos θ: d dθ cos θ + cos2 θ − sin2 θ − sin θ − 2 sin θ cos θ · 1 dx/dθ = · · · = 3(1 + cos θ) (sin θ + 2 sin θ cos θ)2 · 1 −(sin θ + 2 sin θ cos θ) = −3(1 + cos θ) (sin θ + 2 sin θ cos θ)3 . The ellipsis here represents rather a substantial amount of a... |
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EXAMPLE 10.5.2 Find the area under one arch of the cycloid x = t−sin t, y = 1−cos t. We would like to compute Z 2π 0 y dx, but we do not know y in terms of x. However, the parametric equations allow us to make a substitution: use y = 1 − cos t to replace y, and compute dx = (1 − cos t) dt. Then the integral becomes Z ... |
........................................................................................................................................................................................................................................................................................................................ f (x) =... |
ometric series, that is, ∞X n=1 1 2n = 1. It is not hard to see that the following theorem follows from theorem 11.1.2. THEOREM 11.2.2 constant. Then X Suppose that P an and P bn are convergent series, and c is a X X 1. can is convergent and can = c an 264 Chapter 11 Sequences and Series X X X X 2. (an + bn) is converg... |
√ n is a p-series with p = 1/2 < 1, it Since it is typically difficult to compute the value of a series exactly, a good approximation is frequently required. In a real sense, a good approximation is only as good as we know it is, that is, while an approximation may in fact be good, it is only valuable in practice if we c... |
vergence. In the case of the integral test, a single calculation will confirm whichever is the case. To use the comparison test we must first have a good idea as to convergence or divergence and pick the sequence for comparison accordingly. EXAMPLE 11.5.3 Does ∞X n=2 1√ n2 − 3 converge? We observe that the −3 should have... |
specified constant, but we could just as well think of it as a variable, in which case the 11.8 Power Series 281 series ∞X n=0 kxn is a function, namely, the function k/(1 − x), as long as |x| < 1. While k/(1 − x) is a reakxn does have its attractions: sonably easy function to deal with, the more complicated it appears ... |
for x sin(−x). To get from sin x to x sin(−x) we substitute −x for x and then multiply by x. We can do the same thing to the series for sin x: (−1)n (−x)2n+1 (2n + 1)! = x ∞X ∞X x n=0 n=0 (−1)n(−1)2n+1 x2n+1 (2n + 1)! = ∞X n=0 (−1)n+1 x2n+2 (2n + 1)! . As we have seen, a general power series can be centered at a point ... |
ro as N approaches infinity. 11.12 Additional exercises These problems require the techniques of this chapter, and are in no particular order. Some problems may be done in more than one way. Determine whether the series converges. 1. 2. 3. n n2 + n2 + 4)2 ∞X n=0 1 1 · 2 ∞X n=0 ∞X 4. ⇒ n=0 n! 8n 5 5. 1 − 3 4 8 ∞X 1√ + 6.... |
sent a vector as an arrow; the direction of the vector, naturally, is the direction in which the arrow points; the magnitude of the vector is reflected in the length of the arrow. It turns out that many, many quantities behave as vectors, e.g., displacement, velocity, acceleration, force. Already we can get some idea of... |
turn out to be quite useful: i = ⟨1, 0, 0⟩, j = ⟨0, 1, 0⟩, and k = ⟨0, 0, 1⟩. These play much the same role for vectors that the axes play for points. In particular, notice that ⟨v1, v2, v3⟩ = ⟨v1, 0, 0⟩ + ⟨0, v2, 0⟩ + ⟨0, 0, v3⟩ = v1⟨1, 0, 0⟩ + v2⟨0, 1, 0⟩ + v3⟨0, 0, 1⟩ = v1i + v2j + v3k We will frequently want to pr... |
the scalar projection of A onto B may be positive or negative. If it is negative, it means that the projection vector is anti-parallel to B and that the length of the projection vector is the absolute value of the scalar projection. Of course, you can also compute the length of the projection vector as usual, by applyi... |
ing x, y, z by i, j, k gives us k j i a1 a2 a3 b3 b2 b1 = (a2b3 − b2a3)i − (a1b3 − b1a3)j + (a1b2 − b1a2)k = (a2b3 − b2a3)i + (b1a3 − a1b3)j + (a1b2 − b1a2)k = ⟨a2b3 − b2a3, b1a3 − a1b3, a1b2 − b1a2⟩ = A × B. EXAMPLE 12.4.1 Suppose A = ⟨1, 2, 3⟩, B = ⟨4, 5, 6⟩. Then 2 · 6 − 5 · 3)i + (4 · 3 − 1 · 6)j + (1 · 5 − 4 · 2)k... |
lane. We will frequently need to find an equation for a plane given certain information about the plane. While there may occasionally be slightly shorter ways to get to the desired result, 318 Chapter 12 Three Dimensions Figure 12.5.1 A plane defined via vectors perpendicular to a normal. (AP) it is always possible, and ... |
10. Find an equation of the line through (1, 0, 3) and perpendicular to the plane x + 2y − z = 1. ⇒ 11. Find an equation of the line through the origin and perpendicular to the plane x + y − z = 2. ⇒ 12. Find a and c so that (a, 1, c) is on the line through (0, 2, 3) and (2, 7, 5). ⇒ 13. Explain how to discover the sol... |
os ϕ. To see the substitutions for x and y we now view the same point from above, as shown in the right hand graph. The hypotenuse of the triangle in the right hand graph is r = ρ sin ϕ, so the sides of the triangle, as shown, are x = r cos θ = ρ sin ϕ cos θ and y = r sin θ = ρ sin ϕ sin θ. So the upshot is that to con... |
el is rotating at 1 radian per second. Suppose the wheel lies in the y-z plane with center at the origin, and at time t = 0 the spoke lies along the positive y axis and the bug is at the origin. Find a vector function r(t) for the position of the bug at time t. ⇒ 1 − t2, t⟩ 7. What is the difference between the parametr... |
00.50.251.00.50.51.50.751.02.01.01.52.0 Now that we know how to make sense of r′, we immediately know what an antideriva- tive must be, namely 13.2 Calculus with vector functions 337 Z Z Z Z r(t) dt = ⟨ f (t) dt, g(t) dt, h(t) dt⟩, if r = ⟨f (t), g(t), h(t)⟩. What about definite integrals? Suppose that v(t) gives the ve... |
curve but on how quickly we move along the curve. EXAMPLE 13.3.5 Consider r(t) = ⟨cos t, sin t, 0⟩ and s(t) = ⟨cos 2t, sin 2t, 0⟩. Both of these vector functions represent the unit circle in the x-y plane, but if t is interpreted as time, the second describes an object moving twice as fast as the first. Computing the s... |
aN = √ 4 + 36t2 + 36t4 √ 1 + 4t2 + 9t4 . Exercises 13.4. 1. Let r = ⟨cos t, sin t, t⟩. Compute v, a, aT , and aN . ⇒ 2. Let r = ⟨cos t, sin t, t2⟩. Compute v, a, aT , and aN . ⇒ 3. Let r = ⟨cos t, sin t, et⟩. Compute v, a, aT , and aN . ⇒ 4. Let r = ⟨t2, 2t − 3, 3t2 − 3t⟩. Compute v, a, aT , and aN . ⇒ 5. Let r = ⟨et,... |
ays to approach (a, b): along any one of an infinite number of lines, or an infinite number of parabolas, or an infinite number of sine curves, and so on. We might hope that it’s really not so bad—suppose, for example, that along every possible line through (a, b) the value of f (x, y) gets close to L; surely this means t... |
planes. Suppose we want the plane tangent to a surface at a particular point (a, b, c). If we compute the two partial derivatives of the function for that point, we get enough information to determine two lines tangent to the surface, both through (a, b, c) and both tangent to the surface in their 360 Chapter 14 Parti... |
ly) the derivatives of the outside function “times” the derivatives of the inside functions. Not surprisingly, essentially the same chain rule works for functions of more than two variables, for example, given a function of three variables f (x, y, z), where each of x, y and z is a function of t, df dt = ⟨fx, fy, fz⟩ ·... |
allel or anti-parallel, so we want to find the points on the surface with normal parallel or anti-parallel to ⟨3, −1, 3⟩. Let f = x2 + 2y2 + 3z2; the gradient of f is normal to the level surface at every point, so we are looking for a gradient parallel or anti-parallel to ⟨3, −1, 3⟩. The gradient is ⟨2x, 4y, 6z⟩; if it ... |
a minimum. (AP) EXAMPLE 14.7.4 Find all local maxima and minima for f (x, y) = x4 + y4. The derivatives: fx = 4x3 fy = 4y3 fxx = 12x2 fyy = 12y2 fxy = 0. Again there is a single critical point, at (0, 0), and D(0, 0) = fxx(0, 0)fyy(0, 0) − fxy(0, 0)2 = 0 · 0 − 0 = 0, so we get no information. However, in this case it i... |
nt—but there are many such points, in fact an infinite number, as we’ve only shown a few of the level curves. All along the line y = x are points at which two level curves are tangent. While this might seem to be a show-stopper, it is not. Figure 14.8.2 Contour plots for 2x + 2y and xy. 50503001020020y30404010x 14.8 Lag... |
n approximation to the volume under the surface and above the rectangle R = [a, b] × [c, d]. When we take the limit as m and n go to infinity, the double sum becomes the actual volume under the surface, which we divide by (b − a)(d − c) to get the average height. Double sums like this come up in many applications, so in... |
t lines and increases linearly from 2 meters at the south end to 7 meters at the north end. Find the volume of the pool. ⇒ 28. Find the average value of f (x, y) = ey x + ey on the rectangle with vertices (0, 0), (4, 0), √ (4, 1) and (0, 1). ⇒ 29. Figure 15.1.5 shows a temperature map of Colorado. Use the data to estim... |
2)λ in the original integral and still get a finite value for the improper integral? K4K3K2K101234K4K3K2K112345 400 Chapter 15 Multiple Integration 15.3 Moment and Center of Mass Using a single integral we were able to compute the center of mass for a one-dimensional object with variable density, and a two dimensional o... |
dy dx 5x 2 3 dx 3x/2 = = 0 Z 2 0 Z 3x/2 y 5x 2 dx 2 = − 15x2 15x 2 4 − 15x3 12 = 15 − 10 = 5. 0 15x2 4 = 2 0 Pretty much just the way we did for two dimensions we can use triple integration to compute mass, center of mass, and various average quantities. EXAMPLE 15.5.3 Suppose the temperature at a point is given by T =... |
t bounded by the cones ϕ = π/4 and ϕ = 6, and has density proportional to the distance from the √ arctan 2, and the sphere ρ = origin. Find the mass. ⇒ 15.7 Change of Variables One of the most useful techniques for evaluating integrals is substitution, both “u-substitution” and trigonometric substitution, in which we c... |
0⟩, and a small tangent vector, with length approximately equal to the side of the region, is ⟨cos θ0, sin θ0, 0⟩ dr. Likewise, if we fix r = r0 = 0.5, we get the vector function w(θ) = ⟨r0 cos θ, r0 sin θ, 0⟩ with derivative w′(θ) = ⟨−r0 sin θ, r0 cos θ, 0⟩ and a small tangent vector ⟨−r0 sin θ0, r0 cos θ0, 0⟩ dθ when... |
is approximately 2.01.5x1.00.50.0012y340123z456 422 Chapter 16 Vector Calculus ds = |r′| dt = √ 1 + 4t2 dt, so the integral is Z 2 0 p Z 2 p f (t, t2) 1 + 4t2 dt = (t + t2) 1 + 4t2 dt = 0 √ 167 48 17 − 1 12 − 1 64 √ ln(4 + 17). This integral of a function along a curve C is often written in abbreviated form as Z C f (... |
servative. If Py = Qx, then, again provided that F is sufficiently nice, we can be assured that F is conservative. Ultimately, what’s important is that we be able to find f ; as this amounts to finding anti-derivatives, we may not always succeed. EXAMPLE 16.3.3 Find an f so that ⟨3 + 2xy, x2 − 3y2⟩ = ∇f . First, note that ... |
dx + exy dy, where D is described by −2 ≤ x ≤ 2, −1 ≤ y ≤ 1. ⇒ y cos x dx + y sin x dy, where D is described by 0 ≤ x ≤ π/2, 1 ≤ y ≤ 2. ⇒ x2y dx + xy2 dy, where D is described by 0 ≤ x ≤ 1, 0 ≤ y ≤ x. ⇒ √ x y dx + √ x + y dy, where D is described by 1 ≤ x ≤ 2, 2x ≤ y ≤ 4. ⇒ (x/y) dx + (2 + 3x) dy, where D is described... |
he point r(u); as u varies we get a sequence of such circles around the curve r, that is, a tube of radius 1 with r at its center. We can easily change the radius; for example r(u) + ac(u, v) gives the tube radius a; we can make the radius vary as we move along the curve with r(u) + g(u)c(u, v), where g(u) is a functio... |
al, we imagine computing the flux across a very small section of the surface, with area dS, and then adding up all such small fluxes over D with an integral. Suppose that vector N is a unit normal to the surface at a point; F · N is the scalar projection of F onto the direction of N, so it measures how fast the fluid is m... |
his: Z b P a dx dt + Q +R dy dt + ∂z ∂y dy dt ∂z ∂x Z dx dt b P + R dt dx dt ∂z ∂x = = a Z ∂E P + R ∂z ∂x + Q + R ∂z ∂y dt dy dt dy, ∂z ∂y dx + Q + R which now looks just like the line integral of Green’s Theorem, except that the functions P and Q of Green’s Theorem have been replaced by the more complicated P + R(∂z/∂... |
the study of differential equations. 455 456 Chapter 17 Differential Equations 17.1 First Order Differential Equations We start by considering equations in which only the first derivative of the function appears. DEFINITION 17.1.1 A first order differential equation is an equation of the form F (t, y, ˙y) = 0. A solution o... |
geneous linear differential equation is one of the form ˙y + p(t)y = 0 or equivalently ˙y = −p(t)y. “Linear” in this definition indicates that both ˙y and y occur to the first power; “ho- mogeneous” refers to the zero on the right hand side of the first form of the equation. 17.2 First Order Homogeneous Linear Equations 46... |
e tn we need. At each step we do essentially the same calculation, namely (ti+1, yi+1) = (ti + ∆t, yi + ϕ(ti, yi)∆t). We expect that smaller time steps ∆t will give better approximations, but of course it will require more work to compute to a specified time. It is possible to compute a guaranteed 17.4 Approximation 467... |
the solution to the intial value problem ¨y − 4 ˙y + 4y = 0, ˙y(0) = 1. The characteristic polynomial is x2 − 4x + 4 = (x − 2)2, so there y(0) = −3, 472 Chapter 17 Differential Equations y 1 .0 . .. . . . . . . y 0.02 0.01 1 2 . .. . . . . . . 3 4 x 5 0 3 4 x 5 Figure 17.5.1 Graph of a damped oscillation. is one root, ... |
solution to ¨y + 16y = − sin(4t). The roots of the characteristic equation are ±4i, so the solution to the homogeneous equation is A cos(4t) + B sin(4t). Since both cos(4t) and sin(4t) are solutions to the homogeneous equation, C cos(4t) + D sin(4t) is also, so it cannot be a solution to the non-homogeneous equation. ... |
− 20 625 − x2 x21 3.4.5. y = 17x/4 − 41/4 3.4.6. y = 11x/16 − 15/16 3.4.8. y = 19/169 − 5x/338 3.4.9. 13/18 3.5.1. 4x3 − 9x2 + x + 7 3.5.2. 3x2 − 4x + 2/ 3.5.3. 6(x2 + 1)2x p √ x p 3.5.4. 169 − x2 − x2/ p 169 − x2 3.5.5. (2x − 4) 25 − x2− √ (x2 − 4x + 5)x/ 25 − x2 3.5.6. −x/ r2 − x2 p p 1 + x4 1 x(5 − √ x)3/2 3.5.7. 2... |
tance traveled is 6205/98 meters. Z 2π 7.3.2. sin t dt = 0 0 √ 7.3.3. net: 2π, total: 2π/3 + 4 3 7.3.4. 8 7.3.5. 17/3 7.3.6. A = 18, B = 44/3, C = 10/3 8.1.1. −(1 − t)10/10 + C 8.1.2. x5/5 + 2x3/3 + x + C 8.1.3. (x2 + 1)101/202 + C 8.1.4. −3(1 − 5t)2/3/10 + C 8.1.5. (sin4 x)/4 + C 8.1.6. −(100 − x2)3/2/3 + C p 8.1.7. −... |
+ ∞X n=1 ∞X n=1 (2n − 1)! 22n−1(n − 1)! n! xn, R = 1 11.12.21. (−1, 1) 11.12.17. diverges 11.12.18. (−∞, ∞) 11.12.19. (−3, 3) 11.12.20. (−3, 3) 11.12.22. radius is 0—it converges only when 12.2.9. x = 0 √ √ 11.12.23. (− 11.12.24. (−∞, ∞) 3, 3) 11.12.25. 11.12.26. 11.12.27. ∞X n=0 ∞X n=0 ∞X n=0 (ln(2))n n! xn (−1)n n + ... |
.1.24. π/32 15.1.25. 31/8 15.1.26. 128/15 15.1.27. 1800π m3 15.1.28e2 + 8e + 16) 15 1 15 e5/2 15 + √ 15.1.30. 16 − 8 Z 2 Z 9−x2 3 15.1.32. (a) f (x, y) dy dx 0 Z e2 0 Z 2 f (x, y) dx dy Z 0 π/2 ey Z sin x 0 Z Z 4 0 y/4 f (x, y) dy dx f (x, y) dx dy Z 0 3 0 Z √ 9−x2 0 0 f (x, y) dy dx (b) (c) (d) (e) 15.2.1. 4π 15.2.2. ... |
h, 232, 339 arc length parameterization, 340 arccosine, 96 arcsine, 94 area between curves, 191 under a curve, 421 under curve, 151 asymptote, 23, 116 average, 405 average height, 385 B bell curve, 226 binormal, 344, 437 bounded function, 53 C cardioid, 240 Cartesian coordinates, 239, 323 Cauchy Principal Value, 220 ce... |
alent systems is produced by applying row operations. 2.4.1 Row Operations and Equivalent Systems Let A be an m × n matrix and consider the linear system Ax = b. The augmented coefficient matrix of this system is (A|b). The first thing is to point out the role of row operations. What happens when one performs an elementar... |
stency and Uniqueness To finish our treatment of systems (for now), we derive the criteria for consistency and uniqueness. The key concept is the notion of the rank of a matrix, which we saw earlier. Recall that the rank of an m × n matrix A is the number of corners in its reduced row echelon form. 32 Clearly the rank o... |
0 21 4 28 . = = Note how the columns of the product are linear combinations. Computing the product in the opposite order gives a different result: 6 0 −2 · 1 + 7 · 2 −2 · 3 + 7 · 4 6 18 12 22 . = = This example points out that for there exist 2 × 2 matrices A and B such that AB 6= BA, even though both products AB and BA... |
to the original system (by Proposition 2.3). Continuing in this way, we obtain Proposition 3.6. Given a linear system Ax = b, there exists a square matrix B which is a product of elementary matrices, such that the original system is equivalent to Aredx = Bb. Proof. Just apply Propositions 3.5 and 2.3. The advantage of ... |
ich P = S1 · · · Sk. (Recall that a row swap matrix is by definition an elementary matrix obtained by interchanging two rows of In.) Clearly, In is a permutation matrix, and any product of permutation matrices is also a permutation matrix. It remains to see that the inverse of a permutation matrix is also a permutation ... |
iagonal.) Thus, A−1 is a product of lower triangular elementary matrices of type III. But these are elements of Ln, so A−1 is also in Ln. The proof for Un is similar. In fact, one can simply transpose the proof just given. As just noted, every lower triangular unipotent matrix is the product of downward row operations.... |
is therefore unique when a 6= 0 . We also pointed out in the same example that if a = 0, then L and U are not unique, although P and D are. Now consider an n×n system Ax = b. If A is invertible, solving consists of finding A−1. If we write A = LP DU , then A−1 = U −1D−1P −1L−1. In theory, it is simpler to invert each o... |
al numbers. We will begin this chapter with the introduction of the general notion of a field, which generalizes both the reals R and F2. This will immediately give us a whole new way considering of matrices, matrix algebra and, of course, linear systems. Our second goal here is to introduce the notion of an abstract ve... |
sible to solve linear inequalities such as a1x1 + a2x2 + · · · + anxn > c. The reals also have the Archimedean property: if a, b > 0, then there exists an x > 0 so that ax > b. In other words, linear inequalities have solutions. Example 4.3 (F2). The field F2 consisting of 0 and 1 was introduced in the previous chapter.... |
addition and multiplication modulo 4. Then Z4 is a ring, but not a field since 2 · 2 = 0 in Z4 (hence 2 is not invertible). In fact, if q is a composite number, then the ring Zq (defined in an analogous way) is a not a field. Note that the integers Z also form a ring which is not a field. 4.2.1 A Field with Four Elements T... |
ative inverse. The upshot is that we obtain the field C of complex numbers. The marvelous and deep consequence of this definition is that C contains not only numbers such as √ −a, it contains the roots of all polynomial equations with real coefficients. Let us now give the details. The definition of multiplication on R2 is ... |
for all v ∈ V . If v1, . . . , vk ∈ V , then we can define the sum v1 + · · · + vk = k X i=1 vi inductively as (v1+· · ·+vk−1)+vk. Just as we verified for sums in a field, the terms in this sum can be associated in any convenient way, since addition is associative. Similarly, the terms vi can be taken in any order withou... |
t, 108 which enables us (at least partially) to extend our intuition about Rn to C[a, b]. The inner product (f, g) of f, g ∈ C[a, b] is defined by putting (f, g) = Z b a f (t)g(t)dt. The first three axioms for the Euclidean inner product on Rn are verified by applying standard facts about integration proved (or at least s... |
se 4.36. Which of the following subsets of R2 is not a subspace? 115 (a) The line x = y; (b) The unit circle; (c) The line 2x + y = 1; s(d) The first octant x, y ≥ 0. Exercise 4.37. Prove that every line through the origin and plane through the origin in R3 are subspaces. Use this to list all subspaces of R3. Exercise 4... |
r to denote the standard basis vectors in Fn by ei(n) instead of ei . We’ll ignore this point because the context will usually make it clear which Fn we are considering. Since a1 ... an = a1e1 + · · · + anen and In has rank n, it is clear that e1, . . . , en do indeed give a basis of Fn. Example 5.2 (Lines ... |
h a subset exists, since {w1, . . . , wk} has only 2k subsets. It suffices to show v1, . . . , vr are independent, so suppose If ai 6= 0, then a1v1 + · · · + arvr = 0. vi = −1 ai X j6=i ajvj, so if vi is deleted from {v1, . . . , vr}, we still have a spanning set. This contradicts the minimality of r, and hence v1, . . .... |
over F. and compute its dimension. is a subspace of Fn×n. ss ss ss Exercise 5.26. Show that the set of n × n upper triangular real matrices is a subspace of Rn×n. Find a basis and its dimension. Exercise 5.27. Let V be a vector space over Fp of dimensionvn. A linearly independent subset of V with m elements is called ... |
how that the ranks of A, AB and BA are all the same. Exercise 5.31. True or False: rank(A) ≥ rank(A2). Explain your answer. Exercise 5.32. Suppose A and B lie in Fn×n and AB = O. Show that col(B) ⊂ N (A), and conclude that rank(A) + rank(B) ≤ n. Exercise 5.33. Let A be an n × n matrix over the reals R. Which of the fol... |
F. Then we can form a new vector space V × W containing both V and W as subspaces. Definition 5.6. The external direct sum of V and W is the vector space denoted by V × W consisting of all pairs (v, w), where v ∈ V and w ∈ W . Addition is defined component-wise by (v1, w1) + (v2, w2) = (v1 + v2, w1 + w2), and scalar mul... |
tient space V /W is that although the above Theorem tells us its dimension, it doesn’t tell us there is natural choice of a basis. In order to find a basis in the above proof, we first needed a basis of W , which was then extended to a basis of V . Furthermore, the quotient space V /W is an abstract construction. It is n... |
t in Example 6.2 about check digits. If M is a generating matrix, then every element of the linear code C = row(M ) can be expressed as a matrix product of the form (x1 . . . xk)M for a suitable choice of the xi. (To see this, transpose the fact that the column space of M T consists of all vectors of the form M T (y1 .... |
00000 and the following nonzero codewords: (100111), (010101), (001011), (110010), (101100), (011110), (111001). (i) Determine whether or not C is linear. (ii) Compute d(C). (iii) How many elements of C are nearest to (011111)? (iv) Determine whether or not 111111 is a codeword. codeword nearest 111111? If not, is ther... |
only if 2k e X i=0 n i = 2n. This gives the result. 172 Notice that |Be(c)| actually has nothing to do with C. The problem of finding a perfect binary code actually reduces to finding a binary [n, k]code such that |Be(0)| = 2(n−k). If d(C) = 3 or 4, then C is perfect if and only if n = 2n−k − 1, where k = dim C. Some pos... |
Therefore, she has to pass. The other five contestants face similar situations and reason in the same way. With this strategy, their probability of winning the million bucks is 7/8. Can you devise a strategy for how to proceed if there are 4,5 or 6 players? What about 8 or 9? More information about this problem and othe... |
an excellent source of information. Chapter 7 Linear Transformations The purpose of this Chapter is to introduce linear transformations, a way of moving from one vector space to another. In particular, the linear transformations between two vector spaces V and W (over the same field) themselves form a vector space L(V, ... |
the following: (i) φ is injective if and only if there exists a mapping ψ : F (X) → X such that ψ ◦ φ is the identity mapping id : X → X defined by id(x) = x for all x ∈ X. (ii) φ is surjective if and only if there exists a mapping ψ : Y → X such that φ ◦ ψ is the identity mapping id : Y → Y (iii) Conclude that φ is a b... |
ctor with eigenvalue one and any vector orthogonal to a is an eigenvector with eigenvalue zero. 7.3.2 Orthogonal Transformations A transformation T : Rn → Rn is said to be an orthogonal transformation if and only if T (x) · T (y) = x · y (7.4) for all x and y in Rn. By elementary properties of the dot product, it follo... |
nswering this question is to find out how to relate the expansions of a given vector in V with respect to two different bases. 7.4.2 Coordinates With Respect to a Basis Our next goal is to find how one passes from one set of coordinates to another. Let B = {v1, v2, . . . , vn} be a basis of V . Then every v ∈ V has a uniq... |
b ∈ F. In other words, assigning a matrix to a linear transformation is a linear transformation. Exercise 7.41. Let Pn(R) denote the space of polynomials with real coefficients of degree n or less, and let D : Pn(R) → Pn−1(R) be the derivative map. That is, D(f ) = f 0. (i) Show that D is linear, and 212 (ii) find the ma... |
ion S : R2 → R2 defines a C-linear transformation if and only if S = TM , where M has the form M = a −b a b . The astute reader will notice that a −b a M = b = p a2 + b2Rθ, where θ = cos−1(a/ to itself is a multiple of a rotation. √ a2 + b2). Hence, every complex linear map of C = R2 More generally, if S : R2n = Rn × Rn... |
undamental scalar associated to A with a long mathematical history. Most students first encounter it in the statement of Cramer’s Rule, which has as a special case, the formula for the inverse of A (see Section 8.3.3 and Exercise 2.3). The determinant seems to have first appeared in a paper of Liebniz published in 1683, ... |
ix P is a product of row swaps matrices, say P = S1S2 · · · Sm, then det(P ) = (−1)m. Proof. Let P = Pτ , where Pτ = (eτ (1) eτ (2) · · · eτ (n)), and let S = Pσ, where σ is the transposition sending i to j and fixing all other k ∈ Xn. That is, σ = σij. We need to calculate SP . In fact, I claim SP = (eστ (1) eστ (2) · ... |
row successively, we get det(A) = det . Finally, switching the last two rows, we get det(A) = det . Note that switching rows doesn’t change the determinant since −1 = 1 in F2. In fact, we could also deduce that det(A) = 1 using the steps in the previous example. One can simplify evaluating det(A... |
y c21. Expanding the above product gives c21 = −a11 det(A12) + a21 det(A22) − a31 det(A32). 248 But this is exactly the Laplace expansion along the first column for the determinant of the matrix − a11 a11 a13 a21 a21 a23 a31 a31 a33 . The determinant of this matrix is 0, since it has two equal columns. Thus c21 ... |
element of the symmetric group. Hence the definition of the determinant requires some preliminary facts about the symetric group: namely, the definition of the signature of a permutation. If A ∈ Fn×n, then det(A) is an element of F such that det(AB) = det(A) det(B) for all B ∈ Fn×n, det(In) = 1 and det(A) 6= 0 if and onl... |
t F be a field, and suppose V is a finite dimensional vector space over F. Definition 9.1. Suppose T : V → V is a linear map. Then a pair (λ, v), where λ ∈ F and v ∈ V , is called an eigenpair for T if v 6= 0 and T (v) = λv. (9.6) If (λ, v) is an eigenpair for T , we call λ an F-eigenvalue, or, simply, an eigenvalue of T ... |
aracteristic polynomial of a 4 × 4 matrix will involve four 1 × 1 principal minors, six 2 × 2 principal minors, four 3 × 3 principal minors and a single 4 × 4 principal minor. Nevertheless, using Theorem 9.5 is by an effective way to expand det(A − λIn). You should even be able to do the 3 × 3 case without pencil and pa... |
matrix A over F with n distinct eigenvalues in F is diagonalizable. More generally, if V be a finite dimensional vector space over F and T : V → V is a linear transformation with dim V distinct eigenvalues in F, then T is semi-simple. Consider another example. Example 9.11. The counting matrix has characteristic pol... |
O. The above example is an illustration of a nontrivial fact about eigenvalues. Namely, the dimension of the eigenspace of an eigenvalue (i.e. its geometric multiplicity) is at most the algebraic multiplicity of the eigenvalue. This is proved in the next Section. 9.4.3 Tridiagonalization of Complex Matrices We will no... |
Fill in the details of the proof of the Cayley-Hamilton Theorem suggested above using sequences. That is, show that any real or complex matrix is the limit of a sequence of diagonalizable matrices. 291 9.5 The Exponential of a Matrix We now return to the powers of a square matrix A, expanding on the remarks in §9.1. We... |
ace. This is called the least squares problem. It leads naturally to the notions of projections and pseudo-inverses. We will also consider orthonormal bases of both Rn and Cn, which will be also needed in our treatment of the Principal Axis Theorem, proved in Chapter 11. In particular, we will show that every subspace ... |
b. Thus the pseudo-inverse finds the optimal solution of an inconsistent system. The system Ax = b, whether consistent or inconsistent, is solved in the above sense by x = A+b. 10.1.5 Applications of the Pseudoinverse Let’s consider a typical application. Suppose that one has m points (ai, bi) in R2, which represent th... |
/2(1, 1, 1, 1)T and u2 = 1/2(1, −1, −1, 1)T form an orthonormal basis of W , so 311 . 1 1 1 1 −1 −1 1 1 PW = QQT = 1/1 1 −1 1 1 Carrying out the calculation, we find that 1 2 PW = . Here is an important remark that will be expanded in Section 10.4. The formula PW = A(AT A)−1AT only applies when W is a su... |
d y are orthogonal if and only if (x, y) = 0. Thus we can always try to extend the results we obtained for Rn on least squares and projections to an arbitrary V . There is no difficulty in doing this if V is a finite dimensional inner product space. The reason for this is explained in Section 10.4.4 below. Problems arise ... |
y element. A simple, down to earth problem is to describe Rot(S) when S is a Platonic solid in R3. Note that a Platonic solid is a solid whose boundary is a union of plane polygons, all of which are congruent. It has been known since the ancient Greeks that there are exactly five types of Platonic solids: a cube, a regu... |
congruent triangles. How many elements does Rot(S) have? Exercise 10.57. Compute Rot(S) in the following cases: (a) S is the half ball {x2 + y2 + z2 ≤ 1, z ≥ 0}, and (b) S is the solid rectangle {−1 ≤ x ≤ 1, −2 ≤ y ≤ 2, −1 ≤ z ≤ 1}. Exercise 10.58. Suppose H is a reflection of R2. Show that there is a rotation ρ of R3 s... |
e u0 k = uk. Since λ 6= 0, the uj and the 2 and uk are mutually Hermitian orthogonal. Now, let xk = (uk + uk)/ 2. Clearly xk and yk are real; that is, they lie in Rn. yk = i(uk − uk)/ Moreover, the xj and yk are mutually orthogonal, as one can check using the fact that the uj and the uk are mutually Hermitian orthogona... |
rcise 11.17. Suppose B is a real, symmetric 3 × 3 matrix such that (1, 0, 1)T ∈ Null(B − I3), and (1, 1, − 1)T ∈ Null(B − 2I3). If det(B) = 12, find B. Exercise 11.18. Answer either T or F. If T, give a brief reason. If F, give a counter example. (a) The sum and product of two symmetric matrices is symmetric. (b) For an... |
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