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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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