How to show the direct sum of two matrices?

In summary, to show that V is a direct sum of U and W, we must first define the matrices for U and W using the given parameters. Next, we must show that the intersection of U and W is equal to 0, which can be done by equating the elements of U and W and solving for the parameters. Finally, we must demonstrate that any element of V can be written as a sum of an element from U and an element from W.
  • #1
looper
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0

Homework Statement


Let k be a field, V = Mat2x2(k), U:={[a, b], [-b, a] a, b E k} and W:={[a, b], [b, -a] a, b E k}. Show that V is the direct sum of U and W.


Homework Equations





The Attempt at a Solution



Add the matrix for U to the matrix of W. Values in that matrix still exist in k, so V = U + W. Is that the right reasoning? How do you show the intersection of two matrices is 0? Very confused.
 
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  • #2
Write W={[c, d], [d, -c] c, d E k} to make it clear there are four independent parameters here. Equate an element of U with an element of W and show a=0, b=0, c=0 and d=0 from the equations you get. Then you have to show any element of V can be written as a sum of an element from U and an element from W.
 

Related to How to show the direct sum of two matrices?

1. What is the definition of a direct sum of two matrices?

The direct sum of two matrices is a new matrix that is created by combining the two original matrices together in a specific way. It is represented by a diagonal matrix where the elements from the first matrix are on the top left to bottom right diagonal, and the elements from the second matrix are on the bottom left to top right diagonal.

2. How do you show the direct sum of two matrices?

To show the direct sum of two matrices, you first need to ensure that they have the same dimensions. Then, you simply add the elements from the first matrix to the top left to bottom right diagonal of the new matrix, and add the elements from the second matrix to the bottom left to top right diagonal of the new matrix.

3. Can you provide an example of a direct sum of two matrices?

Yes, for example, if we have two matrices A = [1 2; 3 4] and B = [5 6; 7 8], their direct sum would be represented as A⊕B = [1 2 0 0; 3 4 0 0; 0 0 5 6; 0 0 7 8].

4. Is the direct sum of two matrices commutative?

No, the direct sum of two matrices is not commutative. This means that A⊕B is not always equal to B⊕A. In fact, the order in which you add the elements from the two matrices matters in determining the result of the direct sum.

5. Can the direct sum of two matrices be used to perform matrix multiplication?

No, the direct sum of two matrices cannot be used to perform matrix multiplication. It is simply a way to combine two matrices together and does not follow the rules of matrix multiplication. However, it can be useful in other mathematical operations such as finding the determinant or eigenvalues of the combined matrix.

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