**v**_{1 }= (0*,*3*,*−1) *, ***v**_{2 }= (1*,*2*,*0) *,*

and let *W *be the plane spanned by **v**_{1 }and **v**_{2}. Consider the function

*T *: R^{3 }→ R^{3 }where *T*(**x**) = proj_{W }**x ***,*

that is, *T*(**x**) is the projection of **x **onto the plane *W*; **you may assume **that *T *is a linear transformation.

i) Evaluate *T*(5*,*0*,*10).

ii) Find a basis for *W *^{⊥ }in R^{3}.

iii) **Without calculation**, write down the matrix of *T *with respect to the ordered basis {**v**_{1}*,***v**_{2}*,***v**_{3 }}, where **v**_{3 }is the basis element found in part (ii). By drawing a diagram, or otherwise, give reasons for your answer.

iv) Hence or otherwise, find an expression for the matrix of *T *with respect to the standard basis in R^{3}. You may leave your answer as a product of matrices without completing the calculation.

#### Top Answer

i) T(5, 0, 10) = (1, -1, 1) ii) Basis for W are (1, 0, 0), (0, 0, 1) and (0, 0, 1). iii) T ( 1 , 0 , 0 ) = ⎝... View the full answer