How to find the projection

Question

I have the matrix $\mathbf{P} = \boxed{\begin{pmatrix} 4/5 & -2/5 \\ -2/5 & 1/5 \end{pmatrix}}.$

How do I use the matrix $\mathbf{P}$ to figure out $\mathbf{w}_1$ and $\mathbf{w}_2$ in the diagram below

Could someone please show me how they get w2?(or w1)

Answer 1

There is a vector formula for the orthogonal projection onto a vector: if $\langle\cdot,\cdot\rangle$ denotes the inner product, then

$$\text{proj}_u(v)=\frac{\langle u,v\rangle}{\langle u,u\rangle}\,u.$$

Answer 2

You don’t need to compute $P$ explicitly. As the problem states, “use the geometric meaning.”

The first part of the problem asks you what the projection of a vector onto itself is. The answer to that ought to be self-evident. For the second half, notice that $(1,2)$ is orthogonal to $(2,-1)$. Since this is an orthogonal projection, what happens to any vector that’s parallel to the direction in which you’re projecting?

You might also observe that the two diagrams that you’ve included in your question give you the answers!

Answer 3

$P\pmatrix {2\\-1}$ projects the vector onto itself.

Since $\pmatrix {1\\2}$ is orthogonal to $\pmatrix {2\\-1}$

$P\pmatrix {1\\2} = 0$

In terms of the basis $B = \left\{\pmatrix {2\\-1}, \pmatrix {1\\2}\right\}$

$P_B = \pmatrix {1 &0\\0&0}$

How about in terms of the standard basis?

$P = \pmatrix {2&1\\-1&2}\pmatrix {1 &0\\0&0} \pmatrix {2&1\\-1&2}^{-1}$

$P = \pmatrix {\frac {4}{5}&-\frac {2}{5}\\-\frac {2}{5}& \frac {1}{5}}$