Proving the Change of Coordinate Matrix for Left-Multiplication Transformation

[iomtt6076]

Homework Statement

Prove: Let A \in \mathrm{M}_{n \times n}(\mathbb{F}) and let \gamma be an ordered basis for \mathbb{F}^n. Then [\boldmath{L}_A]_{\gamma} = Q^{-1}AQ, where Q is the nxn matrix whose jth column is the jth vector of \gamma.

Homework Equations

\boldmath{L}_A denotes the left-multiplication transformation.

The Attempt at a Solution

Let \beta be the standard ordered basis for \mathbb{F}^n and C the change of coordinate matrix from \beta-coordinates to \gamma-coordinates. Then [\boldmath{L}_A]_{\beta} = A and we have [\boldmath{L}_A]_{\gamma} = C^{-1}AC. Where I'm stuck is showing that the jth column of C is the jth vector of \gamma. Any hints would be appreciated.

 

[Dick]

The matrix C whose jth column is the jth vector of gamma maps the standard basis (1,0...0), (0,1,...0)...(0,0...1) into gamma, doesn't it?

 

[iomtt6076]

The matrix C whose jth column is the jth vector of gamma maps the standard basis (1,0...0), (0,1,...0)...(0,0...1) into gamma, doesn't it?

Dick, thanks for the response. I think I got it now. C should be the change of coordinate matrix from gamma-coordinates to beta-coordinates, not beta to gamma as I had stated above at first. In other words, for [\boldmath{L}_A]_{\gamma} = C^{-1}AC to hold true, C = [\boldmath{I}]_{\gamma}^{\beta} where I is the identity transformation. Then everything makes sense.