# V is vector space of all antisimetric 3x3 matrices

1. Homework Statement

V is vector space of all antisimetric 3x3 matrices. Find the coordinates of the matrix

$$A=$$$$\left| \begin{array}{ccc} \ 0 & 1 & -2 \\ -1 & 0 & -3 \\ 2 & 3 & 0 \end{array} \right|\]$$

relative to the base

$$E_1=$$ $$\left| \begin{array}{ccc} \ 0 & 1 & 1 \\ -1 & 0 & 0 \\ -1 & 0 & 0 \end{array} \right|\]$$

$$E_2=$$$$\left| \begin{array}{ccc} \ 0 & 0 & 1 \\ 0 & 0 & 1 \\ -1 & -1 & 0 \end{array} \right|\]$$

$$E_3=$$$$\left| \begin{array}{ccc} \ 0 & -1 & 0 \\ 1 & 0 & -1 \\ 0 & 1 & 0 \end{array} \right|\]$$

2. Homework Equations

none

3. The Attempt at a Solution

I know to solve just linear map, don't know how to solve this. Please help! Thank you.

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dynamicsolo
Homework Helper
I believe you are looking for the coefficients that work in the linear combination

A = aE_1 + bE_2 + cE_3 .

So you have three unknowns and are going to need three equations. Fortunately, you have three distinct non-zero entries in A to work with. Using the rules of matrix algebra, you'll want to set up equations for those entries of A.

For the A_12 entry, for example, you'll have

1 = a · 1 + b · 0 + c · (-1) .

You would do likewise for, say, the entries A_13 = -2 and
A_23 = -3 . You then have a system of linear equations to solve.

So first he should multiply $$aE_1+bE_2+cE_3=matrixA$$, like this?

dynamicsolo
Homework Helper
So first he should multiply $$aE_1+bE_2+cE_3=matrixA$$, like this?
That's what's meant by a "linear combination": I'm assuming that's what the question is looking for by asking for "coordinates of the matrix" in terms of the basis (I hadn't seen the term used that way before). However, these matrices are simple enough that you can just pick off the entries you need to work with.

Yes, that's it. Thanks guys.
dynamicsolo, I said "the matrix A in ratio (relative) to the basis made of E_1,E_2,E_3"

Yes, that's it. Thanks guys.
dynamicsolo, I said "the matrix A in ratio (relative) to the basis made of E_1,E_2,E_3"