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Suppose I have two subspaces $A,B$ of $\mathbb{R}^n$ whose one basis represented by columns of matrices $A_{n\times m_1},B_{n\times m_2}.[A,B]_{n\times (m_1+m_2)}$ be the appended matrix.

(i) $dim(A+B)+dim(A\cap B)=m_1+m_2$ right?

(ii) $dim(image[A,B])+dim(ker[A,B])=m_1+m_2$ right?

so to get a basis of intersection of two subspace given by linearly indipendent collumn is it enough to find kernel of the appended matrix $[A,B]$? but, $[A,B]:\mathbb{R}^{m_1+m_2}\to\mathbb{R}^n$ so null space will be a subspace of different vector space other than $\mathbb{R}^n$? I have confusion here, whats going on?

(iii) In Matlab is this correct function two get a basis of intersection of two subspace?

Since $[A,B]\begin{pmatrix}x\\y\end{pmatrix}=0\Rightarrow Ax=-By$

function I=Int(A,B)

[m1,n1]=size(A);

[m2,n2]=size(B);

N=null([A,B]);

I=A*N(1:n1,:);

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    The resulting kernel basis vectors in $\mathbb R^{m_1+m_2}$ give you coefficients of linear combinations of the original basis vectors that are in the intersection of the two spaces. – 2017-02-07
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    in (ii) you should have $n$ instead of $m_1+m_2$ on the RHS – 2017-02-18

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