vector space intersection

given the vector spaces $\text{[math]}$ that are subspaces of the vector space $\text{[math]}$ (meaning $\text{[math]}$ ), then the intersection of $\text{[math]}$ and $\text{[math]}$ is defined as:
$\text{[math]}$

$\text{[math]}$

given vector spaces $\text{[math]}$ , to find the intersection $\text{[math]}$ the steps are:

construct the matrix $\text{[math]}$ and find the basis vectors $\text{[math]}$ of its nullspace.
for each basis vector $\text{[math]}$ construct the vector $\text{[math]}$ .
the set $\text{[math]}$ consistute the basis for the intersection space $\text{[math]}$ .

given $\text{[math]}$ , we find the basis and dimension of $\text{[math]}$ :
to find $\text{[math]}$ we need to write $\text{[math]}$ and $\text{[math]}$ as a system of equations

to find the corresponding equations of $\text{[math]}$ we reduce the following matrix:
$\text{[math]}$ and so $\text{[math]}$ , and:
$\text{[math]}$

now to find the corresponding equations of $\text{[math]}$ we reduce the following matrix:
$\text{[math]}$ and so $\text{[math]}$ , and:
$\text{[math]}$

$\text{[math]}$ we put these equations in a matrix and reduce it:
$\text{[math]}$ which gives us:
$\text{[math]}$ and so $\text{[math]}$ and $\text{[math]}$