JSuarez said: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?
Noxide said: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
Yes, it is possible for a square matrix to have equal row and null spaces. This occurs when the rows of the matrix are linearly dependent, meaning that one or more rows can be expressed as a linear combination of the other rows.
No, a non-square matrix cannot have equal row and null spaces. This is because the dimensions of the row and null spaces are determined by the number of rows and columns in the matrix, and a non-square matrix will have different numbers of rows and columns.
When a matrix has equal row and null spaces, it means that the rows of the matrix are linearly dependent and can be expressed as a linear combination of each other. This also means that the null space of the matrix is non-trivial, as it contains more than just the zero vector.
To determine if a matrix has equal row and null spaces, you can use the rank-nullity theorem. This theorem states that the dimension of the row space plus the dimension of the null space equals the number of columns in the matrix. If the dimensions of the row and null spaces are equal, then the matrix has equal row and null spaces.
The row space and null space of a matrix are complementary subspaces. This means that any vector in the null space is orthogonal to any vector in the row space, and vice versa. Additionally, the dimensions of the row and null spaces add up to the number of columns in the matrix.