Let $E\subset \Bbb R^3$ be a plane $\{(x,y,z)^T\in \Bbb R^3: x+y+z=0 \}$. Find a orthonormal basis for E and expand it to a orthonormal basis for $\Bbb R^3$. Calculate the $proj_Ew$ for a vector $w=(w_1,w_2,w_3)^T\in \Bbb R^3$
So for the first part of the problem:
Let $\begin{bmatrix}1\\0\\-1\end{bmatrix}$, and $\begin{bmatrix}0\\1\\-1\end{bmatrix}$ be the basis for the plane. We apply Gram-Schmidt.
I can check the results with calculators online, so I won't bore you with that.
We have $b_1=\begin{bmatrix}-1/2\\1\\-1/2\end{bmatrix}$ $b_2=\frac{1}{\sqrt{3/2}}\begin{bmatrix}-1/2\\1\\-1/2\end{bmatrix}$
Now we take a linearly independent vector relative to the two vectors we used for the plane and apply Gram-Schmidt again. Since this part is not something i want to ask so i will skip the calculations.
So my question is, what is the transpose of a set? What does it do here? My TA said my work is correct, but I just did it the same way i would if E didn't have the transpose sign up there.