1. ## Linear algebra matrices

Let $\displaystyle T: \Re^3 \rightarrow \Re^2$ be a map defined as $\displaystyle T(x_1, x_2, x_3) := (x_1 + x_2, x_1+ x_3)$

a) Find the matrix representation of T.
b) Is T onto? Find the range of T.
c) Is T one-to-one? Find the null set of T.

Thank you!

2. Originally Posted by Linnus
Let $\displaystyle T: \Re^3 \rightarrow \Re^2$ be a map defined as $\displaystyle T(x_1, x_2, x_3) := (x_1 + x_2, x_1+ x_3)$

a) Find the matrix representation of T.
b) Is T onto? Find the range of T.
c) Is T one-to-one? Find the null set of T.
1)$\displaystyle T(1,0,0) = (1,1)$ and $\displaystyle T(0,1,0) = (1,0)$ and $\displaystyle T(0,0,1) = (0,1)$.
Thus the matrix $\displaystyle A$ is $\displaystyle \left[ \begin{array}{ccc}1&1&0\\1&0&1 \end{array}\right]$.

2)If $\displaystyle \bold{b}=(b_1,b_2)$ is in the range it means there is $\displaystyle \bold{x}=(x_1,x_2,x_3)$ so that $\displaystyle A\bold{x}=b$.
This means find all $\displaystyle b_1,b_2$ such that,
$\displaystyle \left[ \begin{array}{cccc}1 & 1 & 0 & b_1 \\ 1 & 0 & 1 & b_2 \end{array} \right]$
Can be reduced to a consistent system.

3)The null set is $\displaystyle A\bold{x} = \bold{0}$ for $\displaystyle \bold{x} = (x_1,x_2,x_3)$.
Thus, you need to solve,
$\displaystyle \left[ \begin{array}{cccc} 1& 1 & 0 & 0 \\ 1& 0 & 1 & 0 \end{array} \right]$