1. ## Matrix proofs

Let $\displaystyle T : \Re \rightarrow \Re ^3$ be given by $\displaystyle T \begin{pmatrix} {x}\\ {y}\\ {z} \end{pmatrix}=\begin{pmatrix} {x}\\ {y}\\ {0} \end{pmatrix}$. This is projection from 3-space onto the xy-plane. Show that T is a linear transformation. What is the matrix associated to T?
I'm not really sure how to prove this is a linear transformation. I couldn't find a definition for a linear transformation either which made the problem more confusing.

I also don't know what it means by "matrix associated to T". Is this all it wants:

$\displaystyle \begin{pmatrix} {1}&{0}&{0}\\ {0}&{1}&{0}\\ {0}&{0}&{0} \end{pmatrix}\begin{pmatrix} {x}\\ {y}\\ {z} \end{pmatrix}=\begin{pmatrix} {x}\\ {y}\\ {0} \end{pmatrix}$

?

Help would be appreciated!

2. Originally Posted by Showcase_22
I'm not really sure how to prove this is a linear transformation. I couldn't find a definition for a linear transformation either which made the problem more confusing.

I also don't know what it means by "matrix associated to T". Is this all it wants:

$\displaystyle \begin{pmatrix} {1}&{0}&{0}\\ {0}&{1}&{0}\\ {0}&{0}&{0} \end{pmatrix}\begin{pmatrix} {x}\\ {y}\\ {z} \end{pmatrix}=\begin{pmatrix} {x}\\ {y}\\ {0} \end{pmatrix}$

?

Help would be appreciated!
Yes!
Since $\displaystyle T: \mathbb{R}^3 \to \mathbb{R}^3$ is a linear transformation it means $\displaystyle T(\bold{x}) = A\bold{x}$.
Where $\displaystyle A$ is a $\displaystyle 3\times 3$ matrix.
That is the associated matrix with $\displaystyle T$.

3. Awesome!

but how do I show it's a linear transformation?

Can I just explain that (x,y,0) is a line since it has only one dimension (length)?

4. Originally Posted by Showcase_22
but how do I show it's a linear transformation?
To show that $\displaystyle T$ is a linear transformation you need to show that: $\displaystyle T(\bold{x}+\bold{y}) = T(\bold{x}) + T(\bold{y})$ and $\displaystyle T(k\bold{x}) = kT(\bold{x})$.
Can you do those two steps?

5. yes I can (and have!)

Cheers! That's some homework I won't have to do at the weekend! =D