Linear Algebra - Concept Question

[ChemistryNat]

Homework Statement

Please See Attached

Homework Equations

The Attempt at a Solution

Since matrix B is an invertible 2x2 matrix, its row reduced echelon form will be the 2x2 identity matrix. Therefore, B, has rowspace span{[1,0][0,1]}, nullspace is the empty set and dimcol(B) is 2

Row reducing the given numerical matrix gives [(1,0),(0,1),(0,0)]
which has rowspace span{[1,0][0,1]}, nullspace is the empty set and columnspace span {[2,-1,0],[2,3,1]}Since the nullspace and rowspace of matrix B and the numerical matrix are equal, is this sufficient to say that the product, matrix A will also have the same nullspace and rowspace?

Since the numerical matrix is a 3x2 and B is 2x2, then A is 3x2
so the columnspace of A must involve column vectors with 3 variables
I'm assuming that the columns of A are linearly dependent on the columns of that explicit matrix and that the columns of A are the columns of the explicit matrix?

thank you!

 

[mighty2000]

Hello,

Thank you for your post. It seems like you have a good understanding of matrix B and its properties. However, to answer your question about matrix A, we need to know more information about it. The size of matrix A will depend on the operation that is being performed between matrix B and the numerical matrix.

If matrix A is the product of matrix B and the numerical matrix, then it will have the same nullspace and rowspace as matrix B. This is because the product of two invertible matrices will also be invertible, and therefore have the same nullspace and rowspace as the original matrix.

However, if matrix A is the sum or difference of matrix B and the numerical matrix, then it may have a different nullspace and rowspace. The nullspace and rowspace of a sum or difference of matrices is not necessarily the same as the individual nullspaces and rowspaces of the matrices.

In summary, if matrix A is the product of matrix B and the numerical matrix, then it will have the same nullspace and rowspace as matrix B. But if matrix A is the sum or difference of matrix B and the numerical matrix, then it may have a different nullspace and rowspace.

I hope this helps clarify things for you. Let me know if you have any further questions. Good luck with your research!