1.

1. Let

   

   Compute both Ax and

   

   and verify that they are equal.

2. Let A ? R ^{n n} and x ? R ⁿ, and suppose the columns of A are

   

   so that the ( i, j)-entry of A is ( v _j) _i. Compute both ( Ax) _i and ( x ₁ v ₁ + x ₂ v ₂ + + x _n v _n) _i, and verify that they are equal.

2.

Is

a basis for R ³? (Hint: As explained in the last paragraphs of this section, the three given vectors form a basis for R ⁿ if and only if Ax = b has a unique solution for every b ? R ⁿ, where A is the 3x3 matrix whose columns are the three given vectors.)

3.

Is

a basis for R ³? (See the hint for the previous exercise.)

4.

Show that

is a basis for , the space of polynomials of degree 2 or less. (Hint: Verify directly that the definition holds.)

5.

Show that { L ₁, L ₂, L ₃}, defined in Example3.28, is a basis for . (Hint: Use (3.9) to show that

holds for every )

6.

Let V be the space of all continuous, complex-valued functions defined on the real line:

Define W to be the subspace of V spanned by