# Thread: Matrix for a projection

1. ## Matrix for a projection

The problem is to find the matrix in the standard basis for the projection R^2 of the projection onto the line 2y=x.

So this means the solution should be a linearly independent set of 2 vectors, but how do I figure this out, is it just a matter of picking arbitrary points for x and y?

2. ## Re: Matrix for a projection

Do you understand what is meant by the projection of $\displaystyle \mathbb{R}^2$ onto a line? That means given a point $\displaystyle (x_0,y_0)$, you find the orthogonal projection onto the line. So, you are going from a two-dimensional vector space to a one-dimensional vector space. That means the solution will be a linearly dependent set of 2 vectors (since the image of the transformation will be one dimension less than the domain).

So, if the transformation is $\displaystyle T$, then $\displaystyle T\begin{bmatrix}x_0 \\ y_0\end{bmatrix} = \begin{bmatrix}2y \\ y\end{bmatrix}$. How do you know that the projection is orthogonal? You take the vector from the starting point to the ending point and make sure it is orthogonal to the vector from the origin to the ending point.

In other words, you want $\displaystyle \begin{bmatrix}2y - x_0 \\ y - y_0\end{bmatrix}\cdot \begin{bmatrix}2y \\ y\end{bmatrix} = 0$ (take the dot product and get zero).