Krdytkyu

I want to prove that
$N(A)$ $intersection$ $R(A)$={$0$} where $A$ be a $m×n$ matrix. & $N(A)$ is the null space of $A$ & $R(A)$ is the row space $A$.
Using their orthogonality we can say that my statement is true but I am trying another method.....
In faCt, I am trying to prove dimension of the intersection of the subspaces is 0.That's all.

If you want to consider the intersection of $N(A)$ and $R(A)$ (row space), you have to define the latter as the column space of $A^T$, because the span of the rows of $A$ lives in a different vector space than $N(A)$.

With that interpretation, let $vin N(A)cap R(A)$. Then there exists $x$ such that $v=A^Tx$. Also $Av=0$, so $AA^Tx=0$; in particular
$$
0=x^TAA^Tx=(A^Tx)^T(A^Tx)=v^Tv
$$
implying $v=0$.

Here $X^T$ denotes the transpose matrix.

Orthogonality is how you prove this, since it's a fundamental definition of the null space of a matrix: the null space is the space that is orthogonal to the row space. We can make some progress with dimensions, but we'll have to return to the orthogonality argument eventually:

If by some reason the two spaces had non trivial intersection, then some nonzero vectors in the null space could be represented as linear combinations of those in the row space. If that were the case, then some sub-space of the row space would be the basis for a sub-space of the null space, and therefore the some nonzero vectors of one are linearly dependent on nonzero vectors in the other. However, there is a theorem that states that orthogonal sets of vectors are always linearly independent. This is a contradiction unless the only vector in the intersection is the zero vector.