16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 3.4.11

Linear Transformations

Mathematics — Linear Algebra · left-multiplication-isomorphism

Problem

Let WW be the space of all n×1n \times 1 column matrices over a field FF. If AA is an n×nn \times n matrix over FF, then AA defines a linear operator LAL_A on WW through left multiplication: LA(X)=AXL_A(X) = AX. Prove that every linear operator on WW is left multiplication by some n×nn \times n matrix, i.e., is LAL_A for some AA. Now suppose VV is an nn-dimensional vector space over the field FF, and let B\mathfrak{B} be an ordered basis for VV. For each α\alpha in VV, define Uα=[α]BU\alpha = [\alpha]_{\mathfrak{B}}. Prove that UU is an isomorphism of VV onto WW. If TT is a linear operator on VV, then UTU−1UTU^{-1} is a linear operator on WW. Accordingly, UTU−1UTU^{-1} is left multiplication by some n×nn \times n matrix AA. What is AA?

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