Question

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Answer 1

We have that the general expression for the projection fo a vector w onto another vector v is the following:

$proj_v(w)=\frac{v\cdot w}{v\cdot v}v$

in this case, we have that v = (0,1,-3), then, if w = (x,y,z), we have that the transformation matrix is the following:

$\begin{gathered} T(w)=proj_v(x,y,z)=\lbrack\frac{(x,y,z)\cdot(0,1,-3)}{(0,1,-3)\cdot(0,1,-3)}\rbrack v=\lbrack\frac{0x+1y-3z}{0+1+9}\rbrack v= \\ \frac{y-3z}{10}(0,-1,3)=(0,-\frac{1}{10}(y-3z),\frac{3}{10}(y-3z)) \end{gathered}$

then, the transformation matrix is T(w) = (0, -1/10 (y-3), 3/10 (y-3z) )

for part b, let w = (1,2,4), then, the transformation matrix of this vector is:

$\begin{gathered} T(1,2,4)=proj_v(1,2,4)=(0.-\frac{1}{10}(2-3(4)),\frac{3}{10}(2-3(4)) \\ =(0,-\frac{1}{10}(-10),\frac{3}{10}(-10))=(0,1,-3) \end{gathered}$

therefore, T(1,2,4) = (0,1,-3)