Determine whether the following sets form subspaces of R2: (a) {(x1, x2) T

Determine whether the following sets form subspaces of R2: (a) {(x1, x2) T | x1 + x2 = 0} (b) {(x1, x2) T | x1x2 = 0} (c) {(x1, x2) T | x1 = 3x2} (d) {(x1, x2) T | |x1|=|x2|} (e) {(x1, x2) T | x2 1 = x2 2}

Determine whether the following sets form subspaces of R3: (a) {(x1, x2, x3) T | x1 + x3 = 1} (b) {(x1, x2, x3) T | x1 = x2 = x3} (c) {(x1, x2, x3) T | x3 = x1 + x2} (d) {(x1, x2, x3) T | x3 = x1 or x3 = x2}

Determine whether the following are subspaces of R22: (a) The set of all 2 2 diagonal matrices (b) The set of all 2 2 triangular matrices (c) The set of all 2 2 lower triangular matrices (d) The set of all 2 2 matrices A such that a12 = 1 (e) The set of all 2 2 matrices B such that b11 = 0 (f) The set of all symmetric 2 2 matrices (g) The set of all singular 2 2 matrices

Determine the null space of each of the following matrices: (a) 2 1 3 2 (b) 1 2 3 1 2 463 (c) 1 3 4 2 1 1 1 3 4 (d) 1 1 1 2 2 2 3 1 1 1 0 5

Determine whether the following are subspaces of P4 (be careful!): (a) The set of polynomials in P4 of even degree (b) The set of all polynomials of degree 3 (c) The set of all polynomials p(x) in P4 such that p(0) = 0 (d) The set of all polynomials in P4 having at least one real root

Determine whether the following are subspaces of C[1, 1]: (a) The set of functions f in C[1, 1] such that f(1) = f(1) (b) The set of odd functions in C[1, 1] (c) The set of continuous nondecreasing functions on [1, 1] (d) The set of functions f in C[1, 1] such that f(1) = 0 and f(1) = 0 (e) The set of functions f in C[1, 1] such that f(1) = 0 or f(1) = 0

Show that Cn[a, b] is a subspace of C[a, b].

Let A be a fixed vector in Rnn and let S be the set of all matrices that commute with A, that is, S = {B | AB = BA} Show that S is a subspace of Rnn

In each of the following determine the subspace of R22 consisting of all matrices that commute with the given matrix: (a) 1 0 0 1 (b) 0 0 1 0 (c) 1 1 0 1 (d) 1 1 1 1

Let A be a particular vector in R22. Determine whether the following are subspaces of R22: (a) S1 = {B R22 | BA = O} (b) S2 = {B R22 | AB = BA} (c) S3 = {B R22 | AB + B = O}

Determine whether the following are spanning sets for R2: (a) 2 1 , 3 2 (b) 2 3 , 4 6 (c) 2 1 , 1 3 , 2 4 (d) 1 2 , 1 2 , 2 4 (e) 1 2 , 1 1

Which of the sets that follow are spanning sets for R3? Justify your answers. (a) {(1, 0, 0)T , (0, 1, 1)T , (1, 0, 1)T } (b) {(1, 0, 0)T , (0, 1, 1)T , (1, 0, 1)T , (1, 2, 3)T } (c) {(2, 1, 2)T , (3, 2, 2)T , (2, 2, 0)T } (d) {(2, 1, 2)T , (2, 1, 2)T , (4, 2, 4)T } (e) {(1, 1, 3)T , (0, 2, 1)T }

Given x1 = 1 2 3 , x2 = 3 4 2 , x = 2 6 6 , y = 9 2 5 (a) Is x Span(x1, x2)? (b) Is y Span(x1, x2)? Prove your answers.

Let A be a 4 3 matrix and let b R4. How many possible solutions could the system Ax = b have if N(A) = {0}? Answer the same question in the case N(A) = {0}. Explain your answers.

Let A be a 4 3 matrix and let c = 2a1 + a2 + a3 (a) If N(A) = {0}, what can you conclude about the solutions to the linear system Ax = c? (b) If N(A) = {0}, how many solutions will the system Ax = c have? Explain.

Let x1 be a particular solution to a system Ax = b and let {z1, z2, z3} be a spanning set for N(A). If Z = z1 z2 z3 , show that y will be a solution to Ax = b if and only if y = x1 + Zc for some c R3.

Let {x1, x2, ... , xk} be a spanning set for a vector space V. (a) If we add another vector, xk+1, to the set, will we still have a spanning set? Explain. (b) If we delete one of the vectors, say, xk, from the set, will we still have a spanning set? Explain.

In R22, let E11 = 1 0 0 0 , E12 = 0 1 0 0 E21 = 0 0 1 0 , E22 = 0 0 0 1 Show that E11, E12, E21, E22 span R22.

Which of the sets that follow are spanning sets for P3? Justify your answers. (a) {1, x2, x2 2} (b) {2, x2, x, 2x + 3} (c) {x + 2, x + 1, x2 1} (d) {x + 2, x2 1}

Let S be the vector space of infinite sequences defined in Exercise 15 of Section 3.1. Let S0 be the set of {an} with the property that an 0 as n . Show that S0 is a subspace of S.

Prove that if S is a subspace of R1, then either S = {0} or S = R1

Let A be an n n matrix. Prove that the following statements are equivalent. (a) N(A) = {0}. (b) A is nonsingular. (c) For each b Rn, the system Ax = b has a unique solution.

Let U and V be subspaces of a vector space W. Prove that their intersection UV is also a subspace of W.

Let S be the subspace of R2 spanned by e1 and let T be the subspace of R2 spanned by e2. Is S T a subspace of R2? Explain.

Let U and V be subspaces of a vector space W. Define U + V = {z | z = u + v where u U and v V} Show that U + V is a subspace of W.

Let S, T, and U be subspaces of a vector space V. We can form new subspaces using the operations of and + defined in Exercises 23 and 25. When we do arithmetic with numbers, we know that the operation of multiplication distributes over the operation of addition in the sense that a(b + c) = ab + ac It is natural to ask whether similar distributive laws hold for the two operations with subspaces. (a) Does the intersection operation for subspaces distribute over the addition operation? That is, does S (T + U) = (S T) + (S U) (b) Does the addition operation for subspaces distribute over the intersection operation? That is, does S + (T U) = (S + T) (S + U)