Let a1 and a2 be linearly independent vectors in R3, and let x be a vector in R2. (a)

In Exercise 1 of Section 3, indicate whether the given vectors form a basis for R2.

In Exercise 2 of Section 3, indicate whether the given vectors form a basis for R3.

Consider the vectors x1 = 2 1 , x2 = 4 3 , x3 = 7 3 (a) Show that x1 and x2 form a basis for R2. (b) Why must x1, x2, and x3 be linearly dependent? (c) What is the dimension of Span(x1, x2, x3)?

Given the vectors x1 = 3 2 4 , x2 = 3 2 4 , x3 = 6 4 8 what is the dimension of Span(x1, x2, x3)?

Let x1 = 2 1 3 , x2 = 3 1 4 , x3 = 2 6 4 (a) Show that x1, x2, and x3 are linearly dependent. (b) Show that x1 and x2 are linearly independent. (c) What is the dimension of Span(x1, x2, x3)? (d) Give a geometric description of Span(x1, x2, x3).

In Exercise 2 of Section 2, some of the sets formed subspaces of R3. In each of these cases, find a basis for the subspace and determine its dimension.

Find a basis for the subspace S of R4 consisting of all vectors of the form (a + b, a b + 2c, b, c)T , where a, b, and c are all real numbers. What is the dimension of S?

Given x1 = (1, 1, 1)T and x2 = (3, 1, 4)T : (a) Do x1 and x2 span R3? Explain. (b) Let x3 be a third vector in R3 and set X = ( x1 x2 x3 ). What condition(s) would X have to satisfy in order for x1, x2, and x3 to form a basis for R3? (c) Find a third vector x3 that will extend the set {x1, x2} to a basis for R3.

Let a1 and a2 be linearly independent vectors in R3, and let x be a vector in R2. (a) Describe geometrically Span(a1, a2). (b) If A = (a1, a2) and b = Ax, then what is the dimension of Span(a1, a2, b)? Explain. 1

The vectors x1 = 1 2 2 , x2 = 2 5 4 , x3 = 1 3 2 , x4 = 2 7 4 , x5 = 1 1 0 span R3. Pare down the set {x1, x2, x3, x4, x5} to form a basis for R3. 1

Let S be the subspace of P3 consisting of all polynomials of the form ax2 + bx + 2a + 3b. Find a basis for S. 1

In Exercise 3 of Section 2, some of the sets formed subspaces of R22. In each of these cases, find a basis for the subspace and determine its dimension. 1

In C[, ], find the dimension of the subspace spanned by 1, cos 2x, and cos2 x. 1

In each of the following, find the dimension of the subspace of P3 spanned by the given vectors: (a) x, x 1, x2 + 1 (b) x, x 1, x2 + 1, x2 1 (c) x2, x2 x 1, x + 1 (d) 2x, x 2 1

Let S be the subspace of P3 consisting of all polynomials p(x) such that p(0) = 0, and let T be the subspace of all polynomials q(x) such that q(1) = 0. Find bases for (a) S (b) T (c) S T 1

In R4, let U be the subspace of all vectors of the form (u1, u2, 0, 0)T , and let V be the subspace of all vectors of the form (0, v2, v3, 0)T . What are the dimensions of U, V, U V, U + V? Find a basis for each of these four subspaces. (See Exercises 20 and 22 of Section 2.) 1

Is it possible to find a pair of two-dimensional subspaces U and V of R3 such that U V = {0}? Prove your answer. Give a geometrical interpretation of your conclusion. [Hint: Let {u1, u2} and {v1, v2} be bases for U and V, respectively. Show that u1, u2, v1, v2 are linearly dependent.] 1

Show that if U and V are subspaces of Rn and U V = {0}, then dim (U + V) = dim U + dim V