1.

1. Show that the basis { v ₁, v ₂, v ₃} from Example 3.33 is an orthonormal basis for R ³.

2. Express the vector

   

   as a linear combination of v ₁, v ₂, v ₃.

2.

Is the basis { L ₁, L ₂, L ₃} for (on the interval [0, 1]) given in Example 3.28 an orthogonal basis?

3.

Let V be an inner product space. Prove that x, y ? V satisfy

if and only if ( x, y) = 0.

4.

Use the results of this section to show that any orthonormal set containing n vectors in R ⁿ is a basis for R ⁿ. (Hint: Since the dimension of R ⁿ is n, it suffices to show either that the orthogonal set spans R ⁿ or that it is linearly independent. Linear independence is probably easier.)

5.

Let W be a subspace of an inner product space V and let { w ₁, w ₂, , w _n} be a basis for W. Show that, for y ? V,

holds if and only if

holds.

6.

Let { w ₁, w ₂, , w _n} be a linearly independent set in an inner product space V, and define G ? R ^{n n} by

Prove that G is...