Dimension of a matrix vectorspace

In summary, the question is asking for the dimension and explicit basis of a vector space of matrices with entries in a field. The dimension would be n*m and the basis would be zero matrices with a 1 in the "ij" entry, starting at 1,1 and ending at n,m.
  • #1
Locoism
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Homework Statement



Let V be the vector space of n × m matrices with entried in a field F . What is the dimension of V ? Give an explicit basis for V over F .

The Attempt at a Solution



The question is a little vague, but if I understand correctly, wouldn't the dimension of V simply be n*m? For the basis (it has m*n elements), would it simply be zero matrices with a 1 in the "ij" entry, starting at 1,1 and ending at n,m?
I feel like the question is just too easy, which is leading me to doubt my answer...
 
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  • #2
That seems correct.
 

1. What is the dimension of a matrix vectorspace?

The dimension of a matrix vectorspace is the number of linearly independent vectors that make up the vectorspace. It is also equal to the number of columns in the matrix.

2. How do you determine the dimension of a matrix vectorspace?

To determine the dimension of a matrix vectorspace, you can use the rank of the matrix. The rank is the number of linearly independent rows or columns in the matrix. The rank is also equal to the number of pivot variables in the matrix.

3. Can the dimension of a matrix vectorspace be greater than the number of rows or columns in the matrix?

No, the dimension of a matrix vectorspace cannot be greater than the number of rows or columns in the matrix. This is because the vectorspace is made up of linear combinations of the rows or columns of the matrix, so it cannot have more dimensions than the matrix itself.

4. How does the dimension of a matrix vectorspace affect its properties?

The dimension of a matrix vectorspace affects its properties in several ways. For example, a vectorspace with a higher dimension can have a larger range of possible values and more complex operations, while a vectorspace with a lower dimension may have more restrictions on its operations. Additionally, a higher dimension can indicate a greater degree of freedom and flexibility in the vectorspace.

5. Can the dimension of a matrix vectorspace change?

Yes, the dimension of a matrix vectorspace can change. If the vectors in the vectorspace are linearly dependent, meaning that they can be expressed as linear combinations of each other, the dimension of the vectorspace will decrease. Similarly, if new linearly independent vectors are added to the vectorspace, the dimension will increase.

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