Linear Algebra With Applications (classic Version)
Linear Algebra With Applications (classic Version)
5th Edition
ISBN: 9780135162972
Author: BRETSCHER, OTTO
Publisher: Pearson Education, Inc.,
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Chapter 5.5, Problem 4E

In ℝ n × m , consider the inner product 〈 A , B 〉 = t r a c e ( A T B ) defined in Example 3.
a. Find a formula for this inner product in ℝ n × 1 = ℝ n .
b. Find a formula for this inner product in ℝ 1 × m (i.e.,the space of row vectors with m components).

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Chapter 5 Solutions

Linear Algebra With Applications (classic Version)

Ch. 5.1 - Find the length of each of the vector vin...Ch. 5.1 - Find the length of each of the vector vin...Ch. 5.1 - Find the length of each of the vector vin...Ch. 5.1 - Find the angle between each of the pairs of...Ch. 5.1 - Find the angle between each of the pairs of...Ch. 5.1 - Find the angle between each of the pairs of...Ch. 5.1 - Prob. 7ECh. 5.1 - Prob. 8ECh. 5.1 - For each pair of vectors u,vlisted in Exercises 7...Ch. 5.1 - For which value(s) the constant k are the vectors...
Ch. 5.1 - Considerthevector u=[131] and v=[100] in n . a....Ch. 5.1 - Give an algebraic proof for thetriangleinequality...Ch. 5.1 - Leg traction. The accompanying figure shows how a...Ch. 5.1 - Leonardo da Vinci and the resolution of forces....Ch. 5.1 - Consider thevector v=[1234] in 4 . Find a basis of...Ch. 5.1 - Consider the vectors...Ch. 5.1 - Find a basis for W , where W=span([1234],[5678]).Ch. 5.1 - Here is an infinite-dimensional version of...Ch. 5.1 - For a line L in 2 , draw a sketch to interpret the...Ch. 5.1 - Refer to Figure 13 of this section. The least-s...Ch. 5.1 - Find scalara, b, c, d, e, f,g such that the...Ch. 5.1 - Consider a basis v1,v2,...,vm of a subspace V of n...Ch. 5.1 - Prove Theorem 5.1 .8d. (V)=V for any subspace V of...Ch. 5.1 - Prob. 24ECh. 5.1 - a. Consider a vector v in n , and a scalar k. Show...Ch. 5.1 - Find the orthogonal projection of [494949] onto...Ch. 5.1 - Find the orthogonal projection of 9e1 onto the...Ch. 5.1 - Find the orthogonal projection of [1000] onto the...Ch. 5.1 - Prob. 29ECh. 5.1 - Consider a subspace V of n and a vector x in n...Ch. 5.1 - Considerthe orthonormal vectors u1,u2,...um , in n...Ch. 5.1 - Consider two vectors v1 and v2 in n . Form the...Ch. 5.1 - Among all the vector in n whose components add up...Ch. 5.1 - Among all the unit vectors in n , find the one for...Ch. 5.1 - Among all the unit vectors u=[xyz] in 3 , find...Ch. 5.1 - There are threeexams in your linear algebra class,...Ch. 5.1 - Consider a plane V in 3 with orthonormal basis...Ch. 5.1 - Consider three unit vectors v1,v2 , and v3 in n ....Ch. 5.1 - Can you find a line L in n and a vector x in n...Ch. 5.1 - In Exercises 40 through 46, consider vectors...Ch. 5.1 - In Exercises 40 through 46, consider vectors...Ch. 5.1 - In Exercises 40 through 46, consider vectors...Ch. 5.1 - In Exercises 40 through 46, consider vectors...Ch. 5.1 - In Exercises 40 through 46, consider vectors...Ch. 5.1 - In Exercises 40 through 46, consider vectors...Ch. 5.1 - In Exercises 40 through 46, consider vectors...Ch. 5.2 - Using paper and pencil, perform the Gram-Schmidt...Ch. 5.2 - Prob. 2ECh. 5.2 - Prob. 3ECh. 5.2 - Using paper and pencil, perform the Gram-Schmidt...Ch. 5.2 - Using paper and pencil, perform the Gram-Schmidt...Ch. 5.2 - Using paper and pencil, perform the Gram-Schmidt...Ch. 5.2 - Prob. 7ECh. 5.2 - Using paper and pencil, perform the Gram-Schmidt...Ch. 5.2 - Prob. 9ECh. 5.2 - Using paper and pencil, perform the Gram-Schmidt...Ch. 5.2 - Using paper and pencil, perform the Gram-Schmidt...Ch. 5.2 - Using paper and pencil, perform the Gram-Schmidt...Ch. 5.2 - Using paper and pencil, perform the Gram-Schmidt...Ch. 5.2 - Using paper and pencil, perform the Gram-Schmidt...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Using paper and pencil, find the QR factorization...Ch. 5.2 - Perform the Gram—Schmidt process on the...Ch. 5.2 - Consider two linearly independent vector v1=[ab]...Ch. 5.2 - Perform the Gram-Schmidt process on the...Ch. 5.2 - Find an orthonormal basis of the plane x1+x2+x3=0Ch. 5.2 - Find an orthonormal basis of the kernel of the...Ch. 5.2 - Find an orthonormal basis of the kernel of the...Ch. 5.2 - Find an orthonormal basis of the kernel of the...Ch. 5.2 - Consider the matrix M=12[111111111111][235046007]...Ch. 5.2 - Consider the matrix...Ch. 5.2 - Find the QR factorization of A=[030000200004] .Ch. 5.2 - Find an orthonormal basis u1,u2,u3 of 3 such that...Ch. 5.2 - Consider an invertible nn matrix A whose...Ch. 5.2 - Consider an invertible upper triangular nn matrix...Ch. 5.2 - The two column vectors v1 and v2 of a 22 matrix...Ch. 5.2 - Consider a block matrix A=[A1A2] with linearly...Ch. 5.2 - Consider an nm matrix A with rank(A)m . Is it...Ch. 5.2 - Consider an nm matrix A with rank(A)=m . Is...Ch. 5.3 - Which of the matrices in Exercises 1 through 4 are...Ch. 5.3 - Which of the matrices in Exercises 1 through 4 are...Ch. 5.3 - Which of the matrices in Exercises 1 through 4 are...Ch. 5.3 - Which of the matrices in Exercises 1 through 4 are...Ch. 5.3 - If the nnmatrices A and B are orthogonal, which of...Ch. 5.3 - If the nnmatrices A and B are orthogonal, which of...Ch. 5.3 - If the nnmatrices A and B are orthogonal, which of...Ch. 5.3 - If the nnmatrices A and B are orthogonal, which of...Ch. 5.3 - If the nnmatrices A and B are orthogonal, which of...Ch. 5.3 - If the nnmatrices A and B are orthogonal, which of...Ch. 5.3 - If the nnmatrices A and B are orthogonal, which of...Ch. 5.3 - If the nnmatrices A and B are symmetric and B is...Ch. 5.3 - If the nnmatrices A and B are symmetric and B is...Ch. 5.3 - If the nnmatrices A and B are symmetric and B is...Ch. 5.3 - If the nnmatrices A and B are symmetric and B is...Ch. 5.3 - If the nnmatrices A and B are symmetric and B is...Ch. 5.3 - If the nnmatrices A and B are symmetric and B is...Ch. 5.3 - If the nnmatrices A and B are symmetric and B is...Ch. 5.3 - If the nnmatrices A and B are symmetric and B is...Ch. 5.3 - IfA andB are arbitrary nnmatrices, which of the...Ch. 5.3 - If A and B are arbitrary nnmatrices, which of the...Ch. 5.3 - If A and B are arbitrary nnmatrices, which of the...Ch. 5.3 - If A and B are arbitrary nnmatrices, which of the...Ch. 5.3 - If A and B are arbitrary nnmatrices, which of the...Ch. 5.3 - If A and B are arbitrary nnmatrices, which of the...Ch. 5.3 - Consider an nn matrix A, a vector v in m , and...Ch. 5.3 - Consider an nn matrix A. Show that A is an...Ch. 5.3 - Show that an orthogonal transformation L from n to...Ch. 5.3 - Consider a linear transformation L from m to n...Ch. 5.3 - Are the rows of an orthogonal matrix A...Ch. 5.3 - a. Consider an nm matrix A such that ATA=Im . Is...Ch. 5.3 - Find all orthogonal 22 matrices.Ch. 5.3 - Find all orthogonal 33 matrices of theform...Ch. 5.3 - Find an orthogonal transformation T form 3 to 3...Ch. 5.3 - Find an orthogonal matrix of the form [2/31/...Ch. 5.3 - Is there an orthogonal transformation T from 3 to...Ch. 5.3 - a. Give an example of a (nonzero) skew-symmetric...Ch. 5.3 - Consider a line L in n , spanned by a unit vector...Ch. 5.3 - Consider the subspace W of 4 spanned by the vector...Ch. 5.3 - Find the matrix A of the orthogonal projection...Ch. 5.3 - Let A be the matrix of an orthogonal projection....Ch. 5.3 - Consider a unit vector u in 3 . We define the...Ch. 5.3 - Consider an nm matrix A. Find...Ch. 5.3 - For which nm matrices A docs theequation...Ch. 5.3 - Consider a QRfactorizationM=QR . Show that R=QTM .Ch. 5.3 - If A=QR is a QR factorization, what is the...Ch. 5.3 - Consider an invertible nn matrix A. Can you write...Ch. 5.3 - Consider an invertible nn matrix A. Can you write...Ch. 5.3 - a. Find all nn matrices that are both orthogonal...Ch. 5.3 - a. Consider the matrix product Q1=Q2S , where both...Ch. 5.3 - Find a basis of the space V of all symmetric 33...Ch. 5.3 - Find a basis of the space V of all skew-symmetric...Ch. 5.3 - Find the dimension of the space of alt...Ch. 5.3 - Find the dimension of the space of all symmetric...Ch. 5.3 - Is the transformation L(A)=AT from 23 to 32...Ch. 5.3 - Is the transformation L(A)=AT from mn to nm...Ch. 5.3 - Find image and kernel of the linear transformation...Ch. 5.3 - Find theimage and kernel of the linear...Ch. 5.3 - Find the matrix of the linear transformation...Ch. 5.3 - Find the matrix of the lineartransformation...Ch. 5.3 - Consider the matrix A=[111325220] with...Ch. 5.3 - Consider a symmetric invertible nn matrix A...Ch. 5.3 - This exercise shows one way to define the...Ch. 5.3 - Find all orthogonal 22 matrices A such that all...Ch. 5.3 - Find an orthogonal 22 matrix A such that all the...Ch. 5.3 - Consider a subspace V of n with a basis v1,...,vm...Ch. 5.3 - The formula A(ATA)1AT for 11w matrix of an...Ch. 5.3 - In 4 , consider the subspace W spanned by the...Ch. 5.3 - In all parts of this problem, let V be the...Ch. 5.3 - An nn matrix A is said to be a Hankel, matrix...Ch. 5.3 - Consider a vector v in n of theform v=[11a2an1]...Ch. 5.3 - Let n be an even positive integer. In both parts...Ch. 5.3 - For any integer m, we define the Fibonacci number...Ch. 5.4 - Consider the subspaceim(A) of 2 , where A=[2436] ....Ch. 5.4 - Consider the subspace im(A) of 3 , where...Ch. 5.4 - Considerasubspace V of n . Let v1,...,vp be a...Ch. 5.4 - Let A bean nm matrix. Is the formula ker(A)=im(AT)...Ch. 5.4 - Let V be the solution space of the linear system...Ch. 5.4 - If A is an nm matrix, is the formula im(A)=im(AAT)...Ch. 5.4 - Consider a symmetric nn matrix A. What is the...Ch. 5.4 - Consider a linear transformation L(x)=Ax from n to...Ch. 5.4 - Consider the linear system Ax=b , where A=[1326]...Ch. 5.4 - Consider a consistent system Ax=b . a. Show that...Ch. 5.4 - Consider a linear transformation L(x)=Ax from n to...Ch. 5.4 - Using Exercise 10 as a guide, define theterm...Ch. 5.4 - Consider a linear transformation L(x)=Ax from n to...Ch. 5.4 - In the accompanying figure, we show the kernel and...Ch. 5.4 - Consider an mn matrix A with ker(A)={0} . Showthat...Ch. 5.4 - Use the formula (imA)=ker(AT) to prove theequation...Ch. 5.4 - Does the equation rank(A)=rank(ATA) hold for all...Ch. 5.4 - Does the equation rank(ATA)=rank(AAT) hold for all...Ch. 5.4 - Find the least-squares solution x* of the system...Ch. 5.4 - By using paper and pencil, find the least-squares...Ch. 5.4 - Find the least-squares solution x* of the system...Ch. 5.4 - Find the least-squares solution x* of the system...Ch. 5.4 - Find the least-squares solution x* of the system...Ch. 5.4 - Find the least-squares solution x* of the system...Ch. 5.4 - Find the least-squares solution x* of the system...Ch. 5.4 - Find the least-squares solution x* of the system...Ch. 5.4 - Consider an inconsistent linear system Ax=b ,...Ch. 5.4 - Consider an orthonormal basis u1,u2,...,un , in n...Ch. 5.4 - Find theleast-squares solution of the system Ax=b...Ch. 5.4 - Fit a linear function of the form f(t)=c0+c1t...Ch. 5.4 - Fit a linear function of the form f(t)=c0+c1t to...Ch. 5.4 - Fit a quadratic polynomial to the data points...Ch. 5.4 - Find the trigonometric function of the form...Ch. 5.4 - Find the function of the form...Ch. 5.4 - Suppose you wish to fit a function of the form...Ch. 5.4 - Let S (t) be the number of daylight hours on the t...Ch. 5.4 - Prob. 37ECh. 5.4 - In the accompanying table, we list the height h,...Ch. 5.4 - In the accompanying table, we list the estimated...Ch. 5.4 - Prob. 40ECh. 5.4 - Prob. 41ECh. 5.4 - Prob. 42ECh. 5.5 - In C[a,b] , define the product f,g=abf(t)g(t)dt ....Ch. 5.5 - Does the equation f,g+h=f,g+f,h hold for all...Ch. 5.5 - Consider a matrix S in nn . In n , define the...Ch. 5.5 - In nm , consider the inner product A,B=trace(ATB)...Ch. 5.5 - Is A,B=trace(ABT) an inner product in nm ?(The...Ch. 5.5 - a. Consider an nm matrix P and an mn matrixQ. Show...Ch. 5.5 - Prob. 7ECh. 5.5 - Prob. 8ECh. 5.5 - Prob. 9ECh. 5.5 - Consider the space P2 with inner product...Ch. 5.5 - Prob. 11ECh. 5.5 - Prob. 12ECh. 5.5 - For a function f in C[,] (with the inner...Ch. 5.5 - Which of the following is an inner product in P2...Ch. 5.5 - Prob. 15ECh. 5.5 - Prob. 16ECh. 5.5 - Prob. 17ECh. 5.5 - Prob. 18ECh. 5.5 - Prob. 19ECh. 5.5 - Prob. 20ECh. 5.5 - Prob. 21ECh. 5.5 - Prob. 22ECh. 5.5 - Prob. 23ECh. 5.5 - Consider the linear space P of all polynomials,...Ch. 5.5 - Prob. 25ECh. 5.5 - Prob. 26ECh. 5.5 - Find the Fourier coefficients of the piecewise...Ch. 5.5 - Prob. 28ECh. 5.5 - Prob. 29ECh. 5.5 - Prob. 30ECh. 5.5 - Gaussian integration,In an introductory...Ch. 5.5 - In the space C[1,1] , we introduce the inner...Ch. 5.5 - a. Let w(t) be a positive-valued function in...Ch. 5.5 - In the space C[1,1] , we define the inner product...Ch. 5.5 - In this exercise, we compare the inner products...Ch. 5 - If T is a linear transformation from n to n...Ch. 5 - If A is an invertible matrix, then the equation...Ch. 5 - Prob. 3ECh. 5 - Prob. 4ECh. 5 - Prob. 5ECh. 5 - Prob. 6ECh. 5 - All nonzero symmetric matrices are invertible.Ch. 5 - Prob. 8ECh. 5 - If u is a unit vector in n , and L=span(u) , then...Ch. 5 - Prob. 10ECh. 5 - Prob. 11ECh. 5 - Prob. 12ECh. 5 - If matrix A is orthogonal, then AT must be...Ch. 5 - If A and B are symmetric nn matrices, then AB...Ch. 5 - Prob. 15ECh. 5 - If A is any matrix with ker(A)={0} , then the...Ch. 5 - If A and B are symmetric nn matrices, then...Ch. 5 - Prob. 18ECh. 5 - Prob. 19ECh. 5 - Prob. 20ECh. 5 - Prob. 21ECh. 5 - Prob. 22ECh. 5 - Prob. 23ECh. 5 - Prob. 24ECh. 5 - Prob. 25ECh. 5 - Prob. 26ECh. 5 - Prob. 27ECh. 5 - If A is a symmetric matrix, vector v is in the...Ch. 5 - The formula ker(A)=ker(ATA) holds for all matrices...Ch. 5 - Prob. 30ECh. 5 - Prob. 31ECh. 5 - Prob. 32ECh. 5 - If A is an invertible matrix such that A1=A , then...Ch. 5 - Prob. 34ECh. 5 - The formula (kerB)=im(BT) holds for all matrices...Ch. 5 - The matrix ATA is symmetric for all matrices A.Ch. 5 - If matrix A is similar to B and A is orthogonal,...Ch. 5 - Prob. 38ECh. 5 - If matrix A is symmetric and matrix S is...Ch. 5 - If A is a square matrix such that ATA=AAT , then...Ch. 5 - Any square matrix can be written as the sum of a...Ch. 5 - If x1,x2,...,xn are any real numbers, then...Ch. 5 - If AAT=A2 for a 22 matrix A, then A must...Ch. 5 - If V is a subspace of n and x is a vector in n ,...Ch. 5 - If A is an nn matrix such that Au=1 for all...Ch. 5 - If A is any symmetric 22 matrix, then there must...Ch. 5 - There exists a basis of 22 that consists of...Ch. 5 - If A=[1221] , then the matrix Q in the QR...Ch. 5 - There exists a linear transformation L from 33 to...Ch. 5 - If a 33 matrix A represents the orthogonal...
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