Finding basis of 3x3 matrix space

In summary, the task is to find a basis for the space of 3x3 matrices with zero row sums and zero column sums. This can be done by considering the condition that each row and column must add up to zero, resulting in a 9 dimensional space with 3 conditions. To find a basis, one can solve for the variables and replace them in the matrix, resulting in 6 basis matrices.
  • #1
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Homework Statement



For my homework assignment, I'm supposed to find a basis for the space of 3x3 matrices that have zero row sums and separately for zero row columns. I am having a hard time with this as it seems to me that there are a lot of combinations I have to consider. For the first, it seems like the rows would have to consist of one 0, one 1, and one -1 in different orders... Is there a better way to do this other than brute force?

Homework Equations





The Attempt at a Solution

 
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  • #2
Well, a general 3 by 3 matrix can be written
[tex]\begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix}[/tex].
The condition that "row sums are 0" means that a+ b+ c= 0, d+ e+ f= 0, and g+ h+ i= 0. Solve that for, say, a, d, and e and replace them in the matrix. Since the space of all 3 by 3 matrices is 9 dimensional, and you have 3 "conditions", you will want 6 basis matrices.
 

What is the basis of a 3x3 matrix space?

The basis of a 3x3 matrix space is a set of linearly independent vectors that can be used to represent any vector in the space through linear combinations. In other words, it is the minimal set of vectors that can span the entire space.

How do you find the basis of a 3x3 matrix space?

To find the basis of a 3x3 matrix space, we can use the Gaussian elimination method to reduce the matrix to its row-echelon form. The non-zero rows in the reduced matrix will form the basis of the space.

Why is finding the basis of a 3x3 matrix space important?

Finding the basis of a 3x3 matrix space is important because it helps us understand the structure and dimensionality of the space. It also allows us to perform computations and transformations on the matrix more efficiently.

Can a 3x3 matrix space have more than one basis?

Yes, a 3x3 matrix space can have multiple bases. This is because there can be more than one set of linearly independent vectors that can span the entire space.

How does the number of basis vectors in a 3x3 matrix space relate to its dimension?

The number of basis vectors in a 3x3 matrix space is equal to its dimension. In this case, since we are dealing with a 3x3 matrix, the dimension of the space is also 3.

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