Let x1, x2, . . . , xk be linearly independent vectors in a vector space V. (a) If we

Determine whether the following vectors are linearly independent in 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

Determine whether the following vectors are linearly independent in R3: (a) 1 0 0 , 0 1 1 , 1 0 1 (b) 1 0 0 , 0 1 1 , 1 0 1 , 1 2 3 (c) 2 1 2 , 3 2 2 , 2 2 0 (d) 2 1 2 , 2 1 2 , 4 2 4 (e) 1 1 3 , 0 2 1

For each of the sets of vectors in Exercise 2, describe geometrically the span of the given vectors.

Determine whether the following vectors are linearly independent in R22: (a) 1 0 1 1 , 0 1 0 0 (b) 1 0 0 1 , 0 1 0 0 , 0 0 1 0 (c) 1 0 0 1 , 0 1 0 0 , 2 3 0 2

Let x1, x2, . . . , xk be linearly independent vectors in a vector space V. (a) If we add a vector xk+1 to the collection, will we still have a linearly independent collection of vectors? Explain. (b) If we delete a vector, say xk , from the collection, will we still have a linearly independent collection of vectors? Explain.

Let x1, x2, and x3 be linearly independent vectors in Rn and let y1 = x1 + x2, y2 = x2 + x3, y3 = x3 + x1 Are y1, y2, and y3 linearly independent? Prove your answer.

Let x1, x2, and x3 be linearly independent vectors in Rn and let y1 = x2 x1, y2 = x3 x2, y3 = x3 x1 Are y1, y2, and y3 linearly independent? Prove your answer.

Determine whether the following vectors are linearly independent in P3: (a) 1, x2, x2 2 (b) 2, x2, x, 2x + 3 (c) x + 2, x + 1, x2 1 (d) x + 2, x2 1

For each of the following, show that the given vectors are linearly independent in C[0, 1]: (a) cos x, sin x (b) x3/2, x5/2 (c) 1, ex + ex , ex ex (d) ex , ex , e2x 1

Determine whether the vectors cos x, 1, sin2(x/2) are linearly independent in C[, ]. 1

Consider the vectors cos(x + ) and sin x in C[, ]. For what values of will the two vectors be linearly dependent? Give a graphical interpretation of your answer. 1

Given the functions 2x and |x|, show that (a) these two vectors are linearly independent in C[1, 1]. (b) the vectors are linearly dependent in C[0, 1]. 1

Prove that any finite set of vectors that contains the zero vector must be linearly dependent. 1

Let v1 and v2 be two vectors in a vector space V. Show that v1 and v2 are linearly dependent if and only if one of the vectors is a scalar multiple of the other. 1

Prove that any nonempty subset of a linearly independent set of vectors {v1, . . . , vn} is also linearly independent. 1

Let A be an m n matrix. Show that if A has linearly independent column vectors, then N(A) = {0}. [Hint: For any x Rn, Ax = x1a1 + x2a2 + + xnan.] 1

Let x1, . . . , xk be linearly independent vectors in Rn, and let A be a nonsingular nn matrix. Define yi = Axi for i = 1, . . . , k. Show that y1, . . . , yk are linearly independent. 1

Let A be a 3 3 matrix and let x1, x2, and x3 be vectors in R3. Show that if the vectors y1 = Ax1, y2 = Ax2, y3 = Ax3 are linearly independent, then the matrix A must be nonsingular and the vectors x1, x2, and x3 must be linearly independent. 1

Let {v1, . . . , vn} be a spanning set for the vector space V, and let v be any other vector in V. Show that v, v1, . . . , vn are linearly dependent. 2

Let v1, v2, . . . , vn be linearly independent vectors in a vector space V. Show that v2, . . . , vn cannot span V.