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v 1 = (0 , 3 , 1) , v 2 = (1 , 2 , 0) , and let W be the plane spanned by v 1 and v 2 .
v1 = (0,3,−1) , v2 = (1,2,0) ,
and let W be the plane spanned by v1 and v2. Consider the function
T : R3 → R3 where T(x) = projW 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 R3.
iii) Without calculation, write down the matrix of T with respect to the ordered basis {v1,v2,v3 }, where v3 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 R3. You may leave your answer as a product of matrices without completing the calculation.
