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)=(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((x,y,z)\cdot(0,1,-3))/((0,1,-3)\cdot(0,1,-3))\rbrack v=\lbrack(0x+1y-3z)/(0+1+9)\rbrack v= \\ (y-3z)/(10)(0,-1,3)=(0,-(1)/(10)(y-3z),(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.-(1)/(10)(2-3(4)),(3)/(10)(2-3(4)) \\ =(0,-(1)/(10)(-10),(3)/(10)(-10))=(0,1,-3) \end{gathered}`

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