## Week 4 problem set for Discussion Board

As we are given $$T = \bigg(\left[\begin{array} {rrr} 2 \\ 1 \end{array} \right] \bigg)$$

and

$$T = \bigg(\left[\begin{array} {rrr} 1 \\ 1 \end{array} \right] \bigg)$$,

we can use the idea of the additive property $$T(a + b) = T(a) + T(b)$$ and add a 3rd

vector c such that $$a + b + c = \left[\begin{array} {rrr} 4 \\ 3 \end{array} \right]$$

with $$a =\left[\begin{array} {rrr} 2 \\ 1 \end{array} \right]$$

and

$$b =\left[\begin{array} {rrr} 1 \\ 1 \end{array} \right]$$

looking at the equation for $$a + b + c$$,

$$c =\left[\begin{array} {rrr} 1 \\ 1 \end{array} \right]$$ that is $$c=b$$

With the new information, we have

$$T(a + b + c) =$$ $$T \bigg(\left[\begin{array} {rrr} 4 \\ 3 \end{array} \right] \bigg) =$$ $$T \bigg(\left[\begin{array} {rrr} 2 \\ 1 \end{array} \right] + 2 \left[\begin{array} {rrr} 1 \\ 1 \end{array} \right]\bigg) =$$

$$T \bigg(\left[\begin{array} {rrr} 2 \\ 1 \end{array} \right] \bigg) + 2 T \bigg(\left[\begin{array} {rrr} 1 \\ 1 \end{array} \right] \bigg) = \left[\begin{array} {rrr} 3 \\ 4 \end{array} \right] + \left[\begin{array} {rrr} -2 \\ 4 \end{array} \right] = \left[\begin{array} {rrr} 1 \\ 8 \end{array} \right]$$