# Thread: Projection matrices in R3

1. ## Projection matrices in R3

The question
a) Find the standard matrix $\displaystyle A_3$ of the transformation which projects $\displaystyle R^3$ onto the x - z plane.
b) Find the standard matrix $\displaystyle A_4$ of the transformation which projects the x - z plane in $\displaystyle R^3$ onto $\displaystyle R^2$.
c) Find the standard matrix $\displaystyle A_5$ of the transformation which 'embeds' $\displaystyle R^2$ as the x - z plane in $\displaystyle R^3$.

I'm not sure how to attempt these. Does it have something to do with $\displaystyle S=GG^T$? I can work out the first one in my head, but I'd like to know the algorithm for solving the questions.

Any assistance would be truly appreciated.

2. ## Re: Projection matrices in R3

Originally Posted by Glitch
a) Find the standard matrix $\displaystyle A_3$ of the transformation which projects $\displaystyle R^3$ onto the x - z plane.
There are several general methods, but in this particular case, the projection of $\displaystyle (x,y,z)$ onto the $\displaystyle xz$ plane directly is $\displaystyle (x',y',z')=(x,0,z)$ that is, $\displaystyle \begin{bmatrix}{x'}\\{y'}\\{z'} \end{bmatrix}=\begin{bmatrix}{1}&{0}&{0}\\{0}&{0}& {0}\\{0}&{0}&{1} \end{bmatrix}\begin{bmatrix}{x}\\{y}\\{z} \end{bmatrix}$ i.e. $\displaystyle A_3=\begin{bmatrix}{1}&{0}&{0}\\{0}&{0}&{0}\\{0}&{ 0}&{1} \end{bmatrix}$

3. ## Re: Projection matrices in R3

Thanks, I worked that much out. It's b) and c) that are giving me the most trouble.

4. ## Re: Projection matrices in R3

b) $\displaystyle (x,y,z)\to (x',y')=(x,z)$ so $\displaystyle \begin{bmatrix}{x'}\\{y'} \end{bmatrix}=\begin{bmatrix}{1}&{0}&{0}\\{0}&{0}& {1} \end{bmatrix}\begin{bmatrix}{x}\\{y}\\{z} \end{bmatrix}$ . Then $\displaystyle A_4=\begin{bmatrix}{1}&{0}&{0}\\{0}&{0}&{1} \end{bmatrix}$