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Invertible Matrices and Orthonormal Basis

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1. (10 pts) Prove or disprove: Let {v1, . . . , vn} be a basis for Rn. If A is an invertible
n × n matrix, then {Av1, . . ., Avn} is also a basis for Rn.

2. (20 pts) In P3, define the function
.....
(a) Show that p, q is an inner product.
(b) Find an orthonormal basis for P3.
(c) Express 7x2 − 2 as a linear combination of the orthonormal basis.

3. (10 pts) Prove or disprove: Let V be a vector space with basis {v1, v2, v3, v4}.
If A = Span(v1, v2) and B = Span(v3, v4), then V = A ⊕ B.

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1. The answer is True.
Proof: Since is a basis of , then we have the following fact.

Now for the invertible matrix , we have . Thus

Thus are ...

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