How do I write vector y as the sum of a vector in each subspace?
I confirmed that the subspaces are orthogonal and any two vectors out of the 4 are orthogonal to each other.
So then I did this:
$$\frac{y\cdot v_1}{\left(v_1\right)^2}\left[v_1\right]+\frac{y\cdot \:v_3}{\left(v_3\right)^2}\left[v_3\right]$$
Which is:
$$\frac{8}{3}\left(1,0,1,1\right)+-\frac{2}{44}\left(1,0,-1,0\right)$$
So I add them and get:
$$\left(\frac{173}{66},0,\frac{179}{66},0\right)$$
Which is the completely wrong answer:
I'm not sure how they got that answer. Any ideas? Thanks in advance.
Best Answer
I'm not sure where you got
$$\frac{y\cdot v_1}{\left(v_1\right)^2}\left[v_1\right]+\frac{y\cdot \:v_3}{\left(v_3\right)^2}\left[v_3\right]$$
from.
You should calculate
$$w_1 = \frac{y\cdot v_1}{\left(v_1\right)^2}\left[v_1\right]+\frac{y\cdot \:v_2}{\left(v_2\right)^2}\left[v_2\right] \in W_1$$
and
$$w_2 = \frac{y\cdot v_3}{\left(v_3\right)^2}\left[v_3\right]+\frac{y\cdot \:v_4}{\left(v_4\right)^2}\left[v_4\right] \in W_2$$
and then you should have $y = w_1 + w_2$.
What you are doing is finding the projection of $y$ onto $W_1$ and $W_2$. And when you have an orthorgonal basis for a subspace with $n$ vectors, you compute the projection onto that subspace by summing $n$ terms that look like $\frac{y\cdot v_i}{\left(v_i\right)^2}\left[v_i\right]$.