- #1
jumbogala
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Homework Statement
Suppose N is an invertible n x n matrix, and let D = {f1, f2, ... , fn} where fi is column i of N for each i. If B is the standard basis of Rn, show that MBD(1Rn) = N.
Call the standard basis of Rn = {E1, ... , En}
Homework Equations
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, f11 is the entry at row 1, column 1. f1 will just denote column 1 of M.
M =
[f11 ... f1n
: :
: :
fn1 ... fnn]
1Rn(f1) = f1, 1Rn(fn) = fn.
f1 can be written as f11E1 + ... + fn1En
fn can be written as fn1E1 + ... + fnnEn
Then MBD(1Rn) 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?