How would I go about finding a projection matrix? As an example, how would I find the projection matrix $\mathbf P$:
$\mathbf{P} \mathbf{v} = \text{The projection of $\mathbf{v}$ onto } \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$
I've no idea as to how to start.
Thanks for the help!
Any $3\times 1$ vector can be expressed as the linear combination of the basis as following $$v=(v_1,v_2,v_3)=v_1(1,0,0)+v_2(0,1,0)+v_3(0,0,1)$$by projection on a fixed vector, we always mean that we preserve the component of any vector to be projected along the given fixed vector i.e. $$Pv=\text{the component of }v\text{ along the vector }(0,1,0)=v_2(0,1,0)$$Here is the desired projection matrix $P$ $$P=\begin{bmatrix}0&0&0\\0&1&0\\0&0&0\end{bmatrix}$$