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linear algebra

Linear Algebra with Applications 7th edition Steven J. Leon - Solutions

If A is an m x n matrix, then AAT and ATA have the same rank. B. If an m x n matrix A has linearly dependent columns and b is a vector in Rm, then b does not have a unique projection onto the column space of A.
If an m x n matrix A has linearly dependent columns and b is a vector in Rm, then b does not have a unique projection onto the column space of A.
If N(A) = (0), then the system Ax = b will have a unique least squares solution.
If Q1 and Q2 are orthogonal matrices, then Q1Q2 is also an orthogonal matrix.
If {u1, u2,..., uk} is an orthonormal set of vectors in Rn and U = (ul, u2, uk) then UTU = Ik (the k × k identity matrix).
Given(a) Find the vector projection p of x onto y. (b) Verify that x - p is orthogonal to p. (c) Verify that the Pythagorean Law holds for x, p, and x - p, that is, ||x|| = ||P||2 + ||x - p||2
GivenIf the Gram-Schmidt process is applied to determine an orthonormal basis for R(A) and a QR factorization of A then, after the first two orthonormal vectors q1, q2 are computed, we have (a) Finish the process. Determine q3 and till in the third columns of Q and R. (b) Use the QR factorization
The functions cos x and sin x are both unit vectors in C[-Ï€, Ï€] with inner product defined by(a) Show that cos x Š¥ sin x. (b) Determine the value of ||cos x + sin x||2.
Consider the vector space C[-1, 1] with inner product defined by(a) Show that form an orthonormal set of vectors. (b) Use the result from part (a) to find the best least squares approximation to h(x) = x1/3 + x2/3 by a linear function.
Let v1 and v2 be vectors in an inner product space V. (a) Is it possible for | < v1, v2 > I > ||v1|| ||v2|| Explain. (b) If | < vl, v2 > | = ||v1|| ||v2|| what can you conclude about the vectors v1 and v2? Explain.
Let v1 and v2 be vectors in an inner product space V. Show that ||v1 + v2|| < (||v1|| + ||v2||)2
Let A be a 7 x 5 matrix with rank equal to 4 and let b be a vector in R8. The four fundamental subspaces associated with A are R(A), N(AT), R(AT), and N(A).(a) What is the dimension of N(AT) and which of the other fundamental subspaces is the orthogonal complement of N(AT)?(b) If x is a vector in
Let x and y be vectors in Rn and let Q be an n x n orthogonal matrix. Show that if z = Qx and w = Qy then the angle between z and w is equal to the angle between x and y.
Let S be the two dimensional subspace of R3 spanned by(a) Find a basis for SŠ¥. (b) Give a geometric description of S and SŠ¥. (c) Determine the projection matrix P that projects vectors in R3 onto SŠ¥.
Given the following table of data pointsfind the best least squares fit by a linear function f(x) = c1 + c2x.
Let {u1, u2, u3} be an orthonormal basis for a three dimensional subspace S of an inner product space V and let x = 2u1 - 2u2 + u3 and y = 3u1 + u2 - 4u3 (a) Determine the value of < x, y >. (b) Determine the value of ||x||.
Let A be a 7 x 5 matrix of rank 4. Let P and Q be the projection matrices that project vectors in R7 onto R(A) and N(AT), respectively. (a) Show that PQ = O. (b) Show that P + Q
Find the angle between the vectors v and w in each of the following. (a) v = (4, l)T, w = (3, 2)T (b) v = (-2, 3, l)T, w = (l,2,4)T
If x = (x1, x2)T, y = (y1, y2)T and z = (z1, z2)T are arbitrary vectors in R2, prove that (a) xTx > 0 (b) xTy = yTx (c) xT(y + z) = xTy + xTz
If u and v are any vectors in R2, show that ||u + v||2 < (||u|| + ||v||)2 and hence l|u + v|| < ||u|| + ||v||. When does equality hold? Give a geometric interpretation of the inequality.
Let x1, x2, x3 be vectors in R3. If x1 ⊥ x2 and x2 ⊥ x3, is it necessarily true that x1 ⊥ x3? Prove your answer.
Let A be a 2 x 2 matrix with linearly independent column vectors a1 and a2. If a1 and a2 are used to form a parallelogram P with altitude h (see the figure), show that(a) h2||a2||2 = l|a1||2||a2||2 - (aT1a2)2(b) Area of P = |det(A)|
Given(a) Determine the angle between x and y. (b) Determine the distance between x and y.
Let x and y be vectors in Rn and define(a) Show that p Š¥ z. Thus p is the vector projection of x onto y; that is, x = p + z, where p and z are orthogonal components of x, and p is a scalar multiple of y. (b) If ||p|| = 6 and ||z|| = 8, determine the value of ||x||.
Use the database matrix Q from Application 1 and search for the key words orthogonality, spaces, vector, only this time give the key word orthogonality twice the weight of the other two key words. Which of the eight modules best matches the search criteria? [Form the search vector using the weights
For each of the following pairs of vectors x and y, find the vector projection p of x onto y and verify that p and x - p are orthogonal. (a) x = (3,5)T. y = (l, l)T (b) x = (2, -5,4)T, y = (1,2.-l)T
Let x and y be linearly independent vectors in R2. If ||x|| =2 and ||y|| = 3, what, if anything, can we conclude about the possible values of |xTy|?
For each of the following matrices, determine a basis for each of the subspaces R(AT), N(A), R(A), and N(AT).(a)(b)
Prove Corollary 5.2.5.
Prove: If A is an m × n matrix and x ∈ Rn, then either Ax = 0 or there exists y ∈ R(AT) such that xTy ≠ 0. Draw a picture similar to Figure 5.2.2 to illustrate this result geometrically for the case where N(A) is a two-dimensional subspace of R3.
Let A be an m x n matrix. Show that (a) If x ∈ N(ATA), then Ax is in both R(A) and N(AT). (b) N(ATA) = N(A). (c) A and ATA have the same rank. (d) If A has linearly independent columns, then ATA is nonsingular.
Let A be an m x n matrix, B an n x r matrix, and C = AB. Show that (a) N(B) is a subspace of N(C). (b) N(C)⊥- is a subspace of N(B)⊥ and, consequently, R(CT) is a subspace of R(BT).
Let U and V be subspaces of a vector space W. If W = U ⊕ V, show that ⋃ ∩ V = (0).
Let A be an m x n matrix of rank r and let |x1,..., xr} be a basis for R(AT). Show that {Ax1,. . . .. . ., Axr] is a basis for R(A).
Let x and y be linearly independent vectors in Rn and let S = Span(x, y). We can use x and y to define a matrix A by settingA = xyT + yxT(a) Show that A is symmetric.(b) Show that N(A) = S⊥.(c) Show that the rank of A must be 2.
Let S be the subspace of R3 spanned by x = (1, - 1, l)T.Give a geometrical description of S and S⊥.
Let S be the subspace of R3 spanned by the vectors x = (x1, x2, x3)T and y (y1,y2, y3)T. LetShow that SŠ¥ = N(A). (b) Find the orthogonal complement of the subspace of R3 spanned by (1, 2, l)T and (1,-1,2)T.
Let aj be a nonzero column vector of an m x n matrix A. Is it possible for aj to be in N(AT)? Explain.
Let S be the subspace of Rn spanned by the vectors x1, x2,..., xk. Show that y ∈ S⊥ if and only if y J ⊥ xi for i = 1 ,. . . . . ., k.
Find the least squares solution to each of the following systems. -x1 + x2 = 10 2x1 + x2 = 5 x1 - 2x2 = 20
Let A be an 8 x 5 matrix of rank 3, and let b be a nonzero vector in N(AT). (a) Show that the system Ax = b must be inconsistent. (b) How many least squares solutions will there be to the system Ax = b? Explain.
Let P = A(ATA)-1 AT, where A is an m × n matrix of rank /;. (a) Show that P2 = P. (b) Prove Pk = P for k = 1,2 (c) Show that P is symmetric. [If B is nonsingular, then (B-1)T = (BT)-1]
Show that ifthen is a least squares solution to the system Ax = b and r is the residual vector.
Let A ∈ Rm×n and let be a solution to the least squares problem Ax = b. Show that a vector y ∈ Rn will also be a solution if and only if y = + z, for some vector z ∈ N(A). [N(ATA) = N(A).]
For each of your solutions in Exercise 1: (a) Determine the projection p = A. (b) Calculate the residual r(). (c) Verify that r() ∈ N(AT).
Find the best least squares fit to the data in Exercise 5 by a quadratic polynomial. Plot the points x = -1, 0, 1, 2 for your function and sketch the graph.
Given a collection of points (x1, y1). (x2 > y2),. . . . . . . . , (xn, yn)> letand let y = c0 + c1x be the linear function that gives the best least squares fit to the points. Show that if = 0 then
The point (,) is the center of mass for the collection of points in Exercise 7. Show that the least squares line must pass through the center of mass. [Use a change of variables z = x - to translate the problem so that the new independent variable has mean 0.]
Let A be an m x n matrix of rank n and let P = A(ATA)-1 AT. (a) Show that Pb - b for every b ∈ R(A). Explain this in terms of projections. (b) If b ∈ R(A)⊥, show that Pb = 0. (c) Give a geometric illustration of parts (a) and (b) if R(A) is a plane through the origin in R3.
If V is an inner product space, show that||v|| = √(v, v)satisfies the first two conditions in the definition of a norm.
Show thatdefines a norm on Rn.
Show thatdefines a norm on Rn.
Let x and y be vectors in an inner product space. Show that if x ⊥ y then the distance between x and y is(||x||2 + ||y||2)1/2
Let x ∈ Rn. Show that ||x||∞ < |x||2
Given x = (1, 1, 1, l)T and y = (8, 2, 2, 0)T:(a) Find the vector projection p of x onto y.(b) Verify that x - p is orthogonal to p.(c) Compute ||x - p||2, ||p||2, ||x||2 and verify that the Pythagorean Law is satisfied.
Let x ∈ R2. Show that ||x||2 < ||x||1. [Write x in the form x1e1 + x2e2 and use the triangle inequality.]
Show that for any u and v in a normed vector space||u + v|| > | ||u|| - ||v|| |
Prove that for any u and v in an inner product space V||u||2 + ||u - v||2 = 2||u||2 + 2||v||2Give a geometric interpretation of this result for the vector space R2
The result of Exercise 24 is not valid for norms other than the norm derived from the inner product. Give an example of this in R2 using || ∙ ||.
Determine whether the following define norms on C[a, b].(a) ||f|| = |f(a)| + |f(b)|(b) |f|| = ˆ«ba |f(x)| dx(c)
Let x ∈ Rn and show that (a) ||x||1 < n||x||∞ (b) ||x||1 < √n||x||∞ Give examples of vectors in Rn for which equality holds in parts (a) and (b).
Sketch the set of points (x1, x2) = xT in R2 such that (a) ||x||2 = l (b) ||x||1 = 1 (c) ||x||∞ = 1
Consider the vector space Rn with inner product (x. y) = xTy. For any n × n matrix A, show that (a) (Ax, y) = (x, ATy) (b) (ATAx, x) = ||Ax||2
Show that equation (2) defines an inner product on Rm×n.
Show that the inner product defined by equation (3) satisfies the last two conditions of the definition of an inner product.
In C[0, 1], with inner product defined by (3), consider the vectors 1 and x. Compute ||1 - p||, ||p||, ||1|| and verify that the Pythagorean Law holds.
In C[-π, π], with inner product defined by (6), show that cos mx and sin nx are orthogonal and that both are unit vectors. Determine the distance between the two vectors.
Write out the Fourier matrix F8. Show that F8P8 can be partitioned zinto block form:
Prove that the transpose of an orthogonal matrix is an orthogonal matrix.
If Q is an n x n orthogonal matrix and x and y are nonzero vectors in Rn, then how does the angle between Qx and Qy compare to the angle between x and y? Prove your answer.
Let Q be an n × n orthogonal matrix. Use mathematical induction to prove each of the following.(a) (Qm)-l = (QT)m = (Qm)T for any positive integer m.(b) ||Qmx|| = ||x|| for any x ∈ Rm.
Let u be a unit vector in Rn and let H = I - 2uuT. Show that H is both orthogonal and symmetric and hence is its own inverse.
Let Q be an orthogonal matrix and let d = det(g). Show that |d| = 1.
Show that the product of two orthogonal matrices is also an orthogonal matrix. Is the product of two permutation matrices a permutation matrix? Explain.
How many n × n permutation matrices are there?
Show that if P is a symmetric permutation matrix then P2k = I and p2k+1 = p.
Show that if U is an n x n orthogonal matrix then u1uT1 + u2uT2 +∙ ∙ ∙ ∙+ unuTn = I
Let(a) Show that {u1, u2, u3} is an orthonormal basis for R3.
Use mathematical induction to show that if Q ∈ Rn×m is both upper triangular and orthogonal, then qj = ±ej j = 1,..., n.
LetShow that the column vectors of A form an orthonormal set in R4.
Let A be the matrix given in Exercise 21. For each of your solutions x to Exercise 21(b), compute Ax and compare it to Pb.
Let A be the matrix given in Exercise 21. Find an orthonormal basis for N(AT).
Let A be an m x n matrix, let P be the projection matrix that projects vectors in Rm onto R(A), and let Q be the projection matrix that projects vectors in R" onto R(AT). Show that (a) I - P is the projection matrix from Rm onto N(AT). (b) I - Q is the projection matrix from Rm onto N(A).
Let P be the projection matrix corresponding to a subspace S of Rm. Show that (a) P2 = P (b) PT = P
Let A be an m x n matrix whose column vectors are mutually orthogonal, and let b ˆˆ Rm. Show that if y is the least squares solution to the system Ax = b then
Consider the inner product space C[0, 1] with inner product defined byLet S be the subspace spanned by the vectors 1 and 2x - 1. (a) Show that 1 and 2x - 1 are orthogonal. (b) Determine ||1|| and ||2x - 1||. (c) Find the best least squares approximation to ˆšx by a function from the
Let S = {l/√2, cos x, cos 2x, ..., cos nx, sin x, sin 2x,..., sin nx} Show that S is an orthonormal set in C[-π, π] with inner product defined by (2).
Find the best least squares approximation to f(x) = |x| on [-π, π] by a trigonometric polynomial of degree less than or equal to 2.
Let {x1, x2,..., xk, xk+1,..., xn} be an orthonormal basis for an inner product space V. Let S1 be the subspace of V spanned by x1,..., xk, and let S2 be the subspace spanned by xk+1, xk+2, ..., xn. Show that S1 ⊥ S2.
Let x be an element of the inner product space V in Exercise 31, and let p1 and p2 be the projections of x onto S1 and S2, respectively. Show that (a) x = P1 + P2. (b) If x ∈ S⊥1, then p1 = 0 and hence S⊥ = S2.
Let S be a subspace of an inner product space V. Let {x1,..., xn} be an orthogonal basis for 5 and let x ˆˆ V. Show that the best least squares approximation to x by elements of S is given by
Let Î¸ be a fixed real number and let(a) Show that {x1, x2) is an orthonormal basis for R2. (b) Verify that c21 + c22 = ||y||2 = y2 + y22
Let u1 and u2 form an orthonormal basis for R2 and let u be a unit vector in R2. If uTu1 = 1/2, determine the value of |uTu2|.
Let (u1, u2, u3} be an orthonormal basis for an inner product space V. If x = c1u1 + c2u2 + c3u3 is a vector with the properties ||x|| = 5, (u1, x) = 4, and x ⊥ u2, then what are the possible values of c1, c2, c3?
The functions cosx and sinx form an orthonormal set in C[-Ï€. Ï€]. Iff(x) = 3 cosx + 2 sin x and g(x) = cosx - sin xuse Corollary 5.5.3 to determine the value of
The setis an orthonormal set of vectors in C[-Ï€, Ï€] with inner product defined by (2). (a) Use trigonometric identities to write the function sin4 x as a linear combination of elements of S. (b) Use part (a) and Theorem 5.5.2 to find the values of the following integrals: (i)
Show that, when carried out in exact arithmetic, the modified Gram-Schmidt process will produce the same orthonormal set as the classical Gram-Schmidt process.
What will happen if the Gram-Schmidt process is applied to a set of vectors {v2, v2, v3}, where v1 and v2 are Jinearly independent, but v3 ∈ Span(v1 v2). Will the process fail? If so how? Explain.
Let A be an m x n matrix of rank n and let be Rm. If Q and R are the matrices derived from applying the Gram-Schmidt process to the column vectors of A and p = c1q1 + c2q2 + --- + cnqn is the projection of b onto R(A), then show that: (a) c = QTb (b) p= QQTb (c) QQT = A{ATA)-1 AT
Let U be an m-dimensional subspace of Rn and let V be a k-dimensional subspace of U, where 0 < k < m.(a) Show that any orthonormal basis (v1, v2,...,vk} for V can be expanded to form an orthonormal basis {v1, v2,..., vk, vk+1,..., vm} for U.(b) Show that if W = Span(v1 +.....+ vk+2 ..., vm),

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