# basis for intersection of two subspaces

Suppose $$\{v_1,v_2,\dots,v_n\}$$ and $$\{w_1,w_2,\dots,w_m\}$$ are bases for an $n$- and $m$-dimensional space respectively.

Form the matrix $$M= \begin{bmatrix} v_1 & v_2 & \ldots & v_n & w_1 & w_2 & \ldots & w_m \end{bmatrix} = \begin{bmatrix} V & W \end{bmatrix}$$ and find basis for its nullspace. Let the basis be $$\{u_1,\ldots,u_p\}$$ where each $u_i= \begin{bmatrix} a_i \\ b_i \end{bmatrix}$.
Then $\{Va_i, \dots, Va_p\}$ is a basis for the intersection subspace.

Could anyone tell me what does he mean by each $u_i=...?$ what is $a_i$ and $b_i$?. well Suppose I found a basis for null space of $[V \hspace{0.2cm}W]$ and suppose it is a matrix $P$ (means its columns are basis for null space), could anyone tell me is he saying that $VP$ will give me the basis for the intersection of subspaces generated by columns of matrices $V$ and $W$? I assume that $v_i$ and $w_i$ are column vector. Thank you for help and solution. http://www.ece.iit.edu/pipermail/ece531/2002-March/000025.html

• I have attempted to retype the text from your picture. Perhaps you can check whether it says something along the lines what you wanted. Commented Jun 22, 2017 at 13:17

You have asked about meaning of $u_i$, $a_i$, $b_i$. This seems to be a reasonable interpretation:
If you have a vector from a nullspace of $M$, this means that $$Mu=0.$$ In particular, the vector $u$ has $n+m$ coordinates, we can divide it into the vector $a$ which contains the first $n$ coordinates and $b$ which contain the rest. So you have $$\begin{bmatrix} V & W \end{bmatrix} \begin{bmatrix} a \\ b \end{bmatrix}=0,$$ i.e., $$Va+Wb=0.$$
In particular, this means that $$Va=-Wb.$$ The vector from the last equation is a linear combination of columns of $V$ and at the same time it is a linear combination of columns of $W$. Which means that it belongs to the intersection of the two subspaces.