V is vector space of all antisimetric 3x3 matrices

In summary, the problem asks to find the coordinates of the matrix A in relation to the basis E_1, E_2, E_3. This can be solved by setting up a system of linear equations, using the non-zero entries of A and the linear combination A = aE_1 + bE_2 + cE_3. Alternatively, the solution can be obtained by directly picking off the entries from A.
  • #1
Theofilius
86
0

Homework Statement



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

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

relative to the base

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

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

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

Homework Equations



none

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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  • #2
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.
 
  • #3
So first he should multiply [tex]aE_1+bE_2+cE_3=matrixA[/tex], like this?
 
  • #4
Physicsissuef said:
So first he should multiply [tex]aE_1+bE_2+cE_3=matrixA[/tex], 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.
 
  • #5
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"
 
  • #6
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"
 

1. What is a vector space?

A vector space is a mathematical structure that consists of a set of objects, called vectors, and a set of operations, such as addition and multiplication, that can be performed on those vectors. These operations must follow certain rules, such as closure, commutativity, and associativity.

2. How is a matrix defined as antisymmetric?

A matrix is considered antisymmetric if it is equal to its negative transpose. In other words, the elements on the main diagonal are all equal to zero, and the elements below the main diagonal are the negatives of the corresponding elements above the main diagonal.

3. What is the dimension of V, the vector space of all antisymmetric 3x3 matrices?

The dimension of V is 3. This means that any vector in V can be represented by a linear combination of 3 basis vectors, which are the standard basis vectors for 3x3 matrices.

4. How does the zero matrix fit into the vector space V?

The zero matrix, which is a matrix where all elements are equal to zero, is considered an element of the vector space V. It serves as the identity element for addition and is also the additive inverse of itself.

5. Can any other matrices be considered elements of V?

No, only antisymmetric 3x3 matrices can be considered elements of the vector space V. Any other type of matrix, such as symmetric or skew-symmetric matrices, would not satisfy the criteria for being an element of V.

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