We have given the matrix
$$ A= \begin{pmatrix} 1 &1 \\ 2& -1 \\ -2 & 4 \end{pmatrix}$$
First the question asks find the orthonormal vectors $q_{1}, q_{2}, q_{3}$ such that $q_{1}, q_{2}$ span the column space of $A$?
Then the question asks which of the fundamental subspaces of the matrix $A$ contains $q_{3}$?
I used Gram-Schmidt method to get $q_{1}$ and $q_{2}$ using the bases for this process as the column space of $A$. i.e Basis: $\{(1, 2, -2),(1, -1, 4)\}$. I got $q_{1}= \frac{1}{3}(1, 2, -2)$, $q_{2}=\frac{1}{3}(2, 1, 2)$.
How do I find q3 and what fundamental subspace of the matrix contains it ?
Best Answer
Complete $q_1,q_2$, to a basis in some way for example take $p$ some element of the canonical bases, and run Gram-Schmidt on $q_1,q_2,p$, to find $q_3$.
The third vector $q_3$ will be orthogonal to the space spanned by the columns. This means $$q_3^TA= 0$$
I cannot know how you should call the space of $x$ with $x^T A= 0$, common names include left null space and cokernel. But it is best to check your reference how it is called there as terminology is not completely uniform.