Linear Algebra with Applications (2-Download)
Linear Algebra with Applications (2-Download)
5th Edition
ISBN: 9780321796974
Author: Otto Bretscher
Publisher: PEARSON
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Magnitude
A vector is an element with both direction and magnitude. A vector is represented using an arrow, where the arrow represents the direction of the vector and its length is represented using the magnitude. The set of vectors in space is known as vector spa…
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Chapter 5.1, Problem 6E

Find the angle θ between each of the pairs of vectors u → and v → in Exercises 4 through 6.
6. u → = [ 1 − 1 2 − 2 ] , v → = [ 2 3 4 5 ]

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Chapter 5.1, Problem 5E
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Chapter 5 Solutions

Linear Algebra with Applications (2-Download)

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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