Linear Transformations (1 Viewer)

[RenegadeMx]

stuck/confused about iii and iv

 

[VBN2470]

For (iii), T(v_1) = v_1 and T(v_2) = v_2, since the projection of vectors lying on a plane onto the plane is that vector itself. Taking co-ordinate vectors of T(v_1) and T(v_2) w.r.t. the ordered basis will give (1, 0, 0)^T and (0, 1, 0)^T as the first two columns. Then, since v_3 is orthogonal to the plane, T(v_3) = 0 (projection of normal vector onto plane is the zero vector) and hence the co-ordinate vector of T(v_3) w.r.t. to the basis is (0, 0, 0)^T which is your 3rd column. So your transformation matrix w.r.t. to the ordered basis will be .

For (iv), I think you are required to calculate the change of basis matrix and should get a matrix in the form .

 

[RenegadeMx]

For (iii), T(v_1) = v_1 and T(v_2) = v_2, since the projection of vectors lying on a plane onto the plane is that vector itself. Taking co-ordinate vectors of T(v_1) and T(v_2) w.r.t. the ordered basis will give (1, 0, 0)^T and (0, 1, 0)^T as the first two columns. Then, since v_3 is orthogonal to the plane, T(v_3) = 0 (projection of normal vector onto plane is the zero vector) and hence the co-ordinate vector of T(v_3) w.r.t. to the basis is (0, 0, 0)^T which is your 3rd column. So your transformation matrix w.r.t. to the ordered basis will be .

For (iv), I think you are required to calculate the change of basis matrix and should get a matrix in the form .

wow didnt realise would be that simple, was overthinking it... thx