Projection onto subspace spanned by a single vector Given the vectors $x=(2,2,3)$ and $u_1=(0,3,0)$, vector $v_1$ is the projection of $x$ on the subspace spanned by $u_1$, so $v_1=\alpha u_1$.
Determine the value of $\alpha$.
My attempt
I haven't dealt with a case where the subspace is spanned by only one vector before. My textbook says that I can write the projection as a linear transformation:
$A^TA\alpha=A^Tx$
$A$ is the matrix that has the basis vectors as columns. The solution is $\alpha=(A^TA)^{-1}A^Tx$.
When I use this I get:
$A=\pmatrix{0\\3\\0}$
$A^TA=\pmatrix{0\\3\\0}(0,3,0)=9$
Then $(A^TA)^{-1}=\frac{1}{9}$, right?
So
$\alpha=\frac{1}{9}(0,3,0)(2,2,3)$
I can't multiply two row vectors. I must have done something wrong. Maybe I can't use the above formula.

Hint:
The component of the vector $\mathbf{x}$ in the direction of $\mathbf{u_1}$ is:
$$
\mathbf{v_1}=\frac{\mathbf{x}\cdot \mathbf{u_1}}{|\mathbf{u_1}|^2}\mathbf{u_1}
$$
