Question 4.1.8: Let T: R³ → R² be a linear transformation, and let B be the ...
Let T: R³ → R² be a linear transformation, and let B be the standard basis for R³.
If
T (e_{1} ) =\begin{bmatrix} 1 \\ 1 \end{bmatrix} T (e_{2} ) = \begin{bmatrix} -1 \\ 2 \end{bmatrix} and T (e_{3} ) = \begin{bmatrix} 0 \\ 1 \end{bmatrix}
find T (ν), where
ν = \begin{bmatrix} 1 \\ 3 \\ 2 \end{bmatrix}
To find the image of the vector ν, we first write the vector as a linear combination of the basis vectors. In this case
ν = e_{1} + 3e_{2} + 2e_{3}
Applying T to this linear combination and using the linearity properties of T, we have
T (ν) = T (e_{1} + 3e_{2} + 2e_{3} )
= T(e_{1} ) + 3T (e_{2} ) + 2T (e_{3} )
= \begin{bmatrix} 1 \\ 1 \end{bmatrix} +3 \begin{bmatrix} -1 \\ 2 \end{bmatrix} +2 \begin{bmatrix} 0 \\ 1 \end{bmatrix}
= \begin{bmatrix} -2 \\ 9 \end{bmatrix}