If n*n matrix, can row space ever be equal to null space?

math_maj0r

If n*n matrix, can row space ever be equal to null space?

P.S.: this is NOT a homework question. It's a general question to get the concepts straight in my head.

JSuarez

Suppose v is a vector that belongs to the row and null spaces of A, then:

[tex]v^T = a^{T}A[/tex]

And:

[tex]Av=\bold 0[/tex]

Therefore, [itex]v^{T}v[/itex] must be equal to what? And what does this imply regarding v?

math_maj0r

What does this symbol mean? a^T

JSuarez

The transpose, of course.

Noxide

The nullspace and rowspace of A are complementary subspaces in R^n (i.e. dim of nullspace + dim of rowspace = n and the dot of any vector in the nullspace with any vector in the rowspace will give 0)

vectors in the basis of the nullspace are of the form Ax=0

vectors in the basis of the rowspace are the pivots rows of the row reduced echelon form of A



Consider the zero Matrix

Noxide

Consider the zero Matrix: it's nullspace will span R^n and it's Rowspace will be the zero vector.

Any invertible matrix has a rowspace that spans R^n and it's nullspace will be the zero vector


Any singular matrix will have k vectors in its rowspace basis and z vectors in its nullspace basis such that z + k = n

since the only vector dotted with itself that gives zero is the zero vector then the nullspace and rowspace would have to both be equal to the zero vector to give zero, meaning that the matrix must be "empty" for that to be possible and an "empty" matrix isn't a matrix at all, so there exists no matrix such that nullspace = rowspace