Introduction to Linear Algebra 4th Edition solutions

Introduction to Linear Algebra 4
Questions 1-10 are about linear independence and linear dependence

Introduction to Linear Algebra 4
Questions 1-10 are about linear independence and linear dependence

Introduction to Linear Algebra 4
Questions 1-10 are about linear independence and linear dependence

Introduction to Linear Algebra 4
Questions 1-10 are about linear independence and linear dependence

Introduction to Linear Algebra 4
Questions 1-10 are about linear independence and linear dependence

Introduction to Linear Algebra 4
Questions 1-10 are about linear independence and linear dependence

Introduction to Linear Algebra 4
Questions 1-10 are about linear independence and linear dependence

Introduction to Linear Algebra 4
Questions 1-10 are about linear independence and linear dependence

Introduction to Linear Algebra 4
Questions 1-10 are about linear independence and linear dependence

Introduction to Linear Algebra 4
Questions 1-10 are about linear independence and linear dependence

Introduction to Linear Algebra 4
Questions 11-15 are about the space spanned by a set of vectors. Take all linear combinations of the vectors.

Introduction to Linear Algebra 4
Questions 11-15 are about the space spanned by a set of vectors. Take all linear combinations of the vectors.

Introduction to Linear Algebra 4
Questions 11-15 are about the space spanned by a set of vectors. Take all linear combinations of the vectors.

Introduction to Linear Algebra 4
Questions 11-15 are about the space spanned by a set of vectors. Take all linear combinations of the vectors.

Introduction to Linear Algebra 4
Questions 15-25 are about the requirements for a basis

Introduction to Linear Algebra 4
Questions 15-25 are about the requirements for a basis

Introduction to Linear Algebra 4
Questions 15-25 are about the requirements for a basis

Introduction to Linear Algebra 4
Questions 15-25 are about the requirements for a basis

Introduction to Linear Algebra 4
Questions 15-25 are about the requirements for a basis

Introduction to Linear Algebra 4
Questions 15-25 are about the requirements for a basis

Introduction to Linear Algebra 4
Questions 15-25 are about the requirements for a basis

Introduction to Linear Algebra 4
Questions 15-25 are about the requirements for a basis

Introduction to Linear Algebra 4
Questions 15-25 are about the requirements for a basis

Introduction to Linear Algebra 4
Questions 15-25 are about the requirements for a basis

Introduction to Linear Algebra 4
Questions 15-25 are about the requirements for a basis

Introduction to Linear Algebra 4
Questions 26-30 are about spaces where the "vectors" are matrices.

Introduction to Linear Algebra 4
Questions 26-30 are about spaces where the "vectors" are matrices.

Introduction to Linear Algebra 4
Questions 26-30 are about spaces where the "vectors" are matrices.

Introduction to Linear Algebra 4
Questions 26-30 are about spaces where the "vectors" are matrices.

Introduction to Linear Algebra 4
Questions 26-30 are about spaces where the "vectors" are matrices.

Introduction to Linear Algebra 4
Questions 31-35 are about spaces where the "vectors" are functions.

Introduction to Linear Algebra 4
Questions 31-35 are about spaces where the "vectors" are functions.

Introduction to Linear Algebra 4
Questions 31-35 are about spaces where the "vectors" are functions.

Introduction to Linear Algebra 4
Questions 31-35 are about spaces where the "vectors" are functions.

Introduction to Linear Algebra 4
Questions 31-35 are about spaces where the "vectors" are functions.

Introduction to Linear Algebra 4
Find a basis for -the space S of vectors (a, b, c, d) with a + c + d = 0 and also for the space T with a + b = 0 and c = 2d. What is the dimension of the intersection SnT?

Introduction to Linear Algebra 4
If AS = SA for the shift matrix S, show that A must have this special form: If [~ ! i] [~ b ~] = [~ b ~] [~ ! i] then A = [~ : g hi 000 000 g h i 00 "The subspace of matrices that commute with the shift S has dimension __

Introduction to Linear Algebra 4
Which of the following are bases for R 3? (a) (1,2,0) and (0, 1,-1) (b) (1,1, -1), (2,3,4), (4,1, -1), (0,1, -1) (c) (1,2,2),(-1,2,1),(0,8,0) (d) (1,2,2),(-1,2,1),(0,8,6)

Introduction to Linear Algebra 4
Suppose A is 5 by 4 with rank 4. Show that Ax = b has no solution when the 5 by 5 matrix [A b] is invertible. Show that Ax = b is solvable when [A b] is singular.

Introduction to Linear Algebra 4
(a) Find a basis for all solutions to d 4 y /dx 4 = y(x). (b) Find a particular solution to d 4 Y / dx4 = Y (x) + 1. Find the complete solution.

Introduction to Linear Algebra 4
Write the 3 by 3 identity matrix as a combination of the other five permutation matrices! Then show that those five matrices are linearly independent. (Assume a combination gives CI PI + ... + Cs Ps = zero matrix, and check entries to prove Ci is zero.) The five permutations are a basis for the subsp

Introduction to Linear Algebra 4
Choose x = (XI,X2,X3,X4) in R4. It has 24 rearrangements like (X2,XI,X3,X4) and (X4, X3, Xl, X2). Those 24 vectors, including x itself, span a subspace S. Find specific vectors x so that the dimension of S is: (a) zero, (b) one, (c) three, (d) four.

Introduction to Linear Algebra 4
Choose x = (XI,X2,X3,X4) in R4. It has 24 rearrangements like (X2,XI,X3,X4) and (X4, X3, Xl, X2). Those 24 vectors, including x itself, span a subspace S. Find specific vectors x so that the dimension of S is: (a) zero, (b) one, (c) three, (d) four.

Introduction to Linear Algebra 4
Mike Artin suggested a neat higher-level proof of that dimension formula in Problem 43. From all inputs V in V and w in W, the "sum transformation" produces v+w. Those outputs fill the space V + W. The nullspace contains all pairs v = u, W = -u for vectors u in V n W. (Then v + W = u - u = 0.) So dim

Introduction to Linear Algebra 4
Inside Rn, suppose dimension (V) + dimension (W) > n. Show that some nonzero vector is in both V and W.

Introduction to Linear Algebra 4
Suppose A is 10 by 10 and A2 = 0 (zero matrix). This means that the column space of A is contained in the, . If A has rank r, those subspaces have dimension r < 10 - r. So the rank IS r < 5. (This problem was added to the second printing: If A2 = 0 it says that r < n/2.)