(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose N is an invertible n x n matrix, and let D = {f_{1}, f_{2}, ... , f_{n}} where f_{i}is column i of N for each i. If B is the standard basis of R^{n}, show that M_{BD}(1_{Rn}) = N.

Call the standard basis of R^{n}= {E_{1}, ... , E_{n}}

2. Relevant equations

3. The attempt at a solution

The first thing I don't get is whether D is a basis. I thought it had to be a basis to do this kind of question, but the problem doesn't specify! I'm going to assume it is...

Now I'm going to write the matrix M, specifying its entries. For example, f_{11}is the entry at row 1, column 1. f_{1}will just denote column 1 of M.

M =

[f_{11}... f_{1n}

: :

: :

f_{n1}... f_{nn}]

1R^{n}(f_{1}) = f_{1}, 1_{Rn}(f_{n}) = f_{n}.

f_{1}can be written as f_{11}E_{1}+ ... + f_{n1}E_{n}

f_{n}can be written as f_{n1}E_{1}+ ... + f_{nn}E_{n}

Then M_{BD}(1_{Rn}) is the coefficients of the above, written in column form, so we get exactly the matrix M.

This seems to prove it! But the question specifies that M is invertible, and I didn't use that fact at all. So I think I may have done something wrong. Can anyone help?

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# Homework Help: Linear Algebra Change Matrices Confusion

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