vector space addition

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

$\text{[math]}$

step 1: prove $\text{[math]}$
$\text{[math]}$

step 2: prove addition closure
let $\text{[math]}$
$\text{[math]}$

step 3: prove multiplication closure
let $\text{[math]}$
$\text{[math]}$

given $\text{[math]}$ , we find the basis and dimension of $\text{[math]}$ :
$\text{[math]}$
we put these vectors in a matrix and reduce it to find out how many linearly independant vectors we have
$\text{[math]}$ which means that only the first 3 vectors are linearly independant so $\text{[math]}$ and $\text{[math]}$