Question

The linear transformation \[T:R^3->R^3\] depicts the vector\[u=(2,3,1)\] in \[T(u)=(4,5,2)\] and the vector \[v=(1,1,2)\] in \[T(v)=(3,-1,2)\] Find \[T(1,3,-4)\]

Answer 1

3,5,2

Answer 2

Could you tell me how you got that answer?

Answer 3

you have already mapped the first two

Answer 4

the third one come from linear interpolation

Answer 5

this is just nomenclaturic bs, right?

Answer 6

It is part of a chapter called: Subspaces and linear transformations, in a book bought to calculus I

Answer 7

OK, gotcha

first \[\left|\begin{matrix}2 & 3 &1 \\ 1 & 1 & 2 \\ 1 & 3 & -4\end{matrix}\right| = 0\]

which makes you wonder

then a bit of row reductions shows that

\[2\left[\begin{matrix}2 \\ 3\\1\end{matrix}\right] - 3\left[\begin{matrix}1 \\ 1\\2\end{matrix}\right] = \left[\begin{matrix}1 \\ 3\\-4\end{matrix}\right]\]

so they are linearly dependent vectors.

Answer 8

so you can apply these rules to find the transformation of the third vector

\(f(\mathbf{x}+\mathbf{y}) = f(\mathbf{x})+f(\mathbf{y}) \!\)
\(f(\alpha \mathbf{x}) = \alpha f(\mathbf{x}) \!\)