Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))
Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))
9th Edition
ISBN: 9780321962218
Author: Steven J. Leon
Publisher: PEARSON
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Chapter 2.2, Problem 1E

Evaluate each of the following determinants by inspection.

(a) | 0 0 3 0 4 1 2 3 1 |

(b) | 1 1 1 3 0 3 1 1 0 0 2 2 1 1 1 2 |

(c) | 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 |

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R denotes the field of real numbers, Q denotes the field of rationals, and Fp denotes the field of p elements given by integers modulo p. You may refer to general results from lectures. Question 1 For each non-negative integer m, let R[x]m denote the vector space consisting of the polynomials in x with coefficients in R and of degree ≤ m. x²+2, V3 = 5. Prove that (V1, V2, V3) is a linearly independent (a) Let vi = x, V2 = list in R[x] 3. (b) Let V1, V2, V3 be as defined in (a). Find a vector v € R[×]3 such that (V1, V2, V3, V4) is a basis of R[x] 3. [8] [6] (c) Prove that the map ƒ from R[x] 2 to R[x]3 given by f(p(x)) = xp(x) — xp(0) is a linear map. [6] (d) Write down the matrix for the map ƒ defined in (c) with respect to the basis (2,2x + 1, x²) of R[x] 2 and the basis (1, x, x², x³) of R[x] 3. [5]
Question 4 (a) The following matrices represent linear maps on R² with respect to an orthonormal basis: = [1/√5 2/√5 [2/√5 -1/√5] " [1/√5 2/√5] A = B = [2/√5 1/√5] 1 C = D = = = [ 1/3/5 2/35] 1/√5 2/√5 -2/√5 1/√5' For each of the matrices A, B, C, D, state whether it represents a self-adjoint linear map, an orthogonal linear map, both, or neither. (b) For the quadratic form q(x, y, z) = y² + 2xy +2yz over R, write down a linear change of variables to u, v, w such that q in these terms is in canonical form for Sylvester's Law of Inertia. [6] [4]
part b please

Chapter 2 Solutions

Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))

Ch. 2.1 - Let A and B be 22 matrics. a) Does...Ch. 2.1 - Let A and B be 22 matrices and let C=[ a 11 a 12 a...Ch. 2.1 - Let A be a symmetric tridiagonal matrix (ie., A is...Ch. 2.2 - Evaluate each of the following determinants by...Ch. 2.2 - Let A=[0123111122331223] (a) Use the elimination...Ch. 2.2 - For each of the following, compute the determinate...Ch. 2.2 - Find all possible choices of c that would make the...Ch. 2.2 - Let Abe an nn matrix and a scalar. Show that...Ch. 2.2 - Let A be a nonsingular matrix. Show that...Ch. 2.2 - Let A and B be 33 matrices with det(A)=4 and...Ch. 2.2 - Show that if E is an elementary matrix, then ET is...Ch. 2.2 - Let E1,E2 , and E3 be 33 elementary matrices of...Ch. 2.2 - Let A and B be row equivalent matrices, and...Ch. 2.2 - Let Abe an nn matrix. Is it possible for A2+I=O in...Ch. 2.2 - Consider the 33 Vandermonde matrix V=[1 x 1 x 1 21...Ch. 2.2 - Suppose that a 33 matrix A factors into a product...Ch. 2.2 - Let A and B be nn matrices. Prove that the product...Ch. 2.2 - Let A and B be nn matrices. Prove that if AB=I ,...Ch. 2.2 - A matrix A is said to be skew symmetric if AT=A ....Ch. 2.2 - Let A be a nonsingular nn matrix with a nonzero...Ch. 2.2 - Let A be a kk matrix and let B be an (nk)(nk)...Ch. 2.2 - Let A and B be kk matrics and let M=[OBAO] Show...Ch. 2.2 - Show that evaluating the determinant of an nn...Ch. 2.2 - Show that the elimination method of computing the...Ch. 2.3 - For each of the following, compute (i) det(A) ,...Ch. 2.3 - Use the Cramer’s rule to solve each of the...Ch. 2.3 - Given A=[121043122] determine the (2, 3) entry of...Ch. 2.3 - Let A be the matrix in Exercise 3. Compute the...Ch. 2.3 - Prob. 5ECh. 2.3 - If A is singular, what can you say about the...Ch. 2.3 - Prob. 7ECh. 2.3 - Prob. 8ECh. 2.3 - Prob. 9ECh. 2.3 - Show that if A is nonsingular, then adj A is...Ch. 2.3 - Prob. 11ECh. 2.3 - Prob. 12ECh. 2.3 - Prob. 13ECh. 2.3 - Prob. 14ECh. 2.3 - Let x, y, and z be vectors in R3 . Show each of...Ch. 2.3 - Let x and y be vectors in R3 and define the skew...Ch. 2 - Generate random 55 matrices with integer entries...Ch. 2 - Prob. 2ECh. 2 - Prob. 3ECh. 2 - Prob. 4ECh. 2 - Prob. 5ECh. 2 - Prob. 6ECh. 2 - Prob. 1CTACh. 2 - For each statement that follows, answer true if...Ch. 2 - For each statement that follows, answer true if...Ch. 2 - Prob. 4CTACh. 2 - Prob. 5CTACh. 2 - For each statement that follows, answer true if...Ch. 2 - For each statement that follows, answer true if...Ch. 2 - Prob. 8CTACh. 2 - Prob. 9CTACh. 2 - Prob. 10CTACh. 2 - Let A and B he 33 matrices with det(A)=4 and...Ch. 2 - Prob. 2CTBCh. 2 - Let A=[1111123413610141020] (a) Compute the LU...Ch. 2 - Prob. 4CTBCh. 2 - Prob. 5CTBCh. 2 - Prob. 6CTBCh. 2 - Let A be an nn matrix and let be a scalar. Show...Ch. 2 - Prob. 8CTBCh. 2 - Prob. 9CTBCh. 2 - Let A be a matrix with integer entries. If...
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