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

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

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