5. [§ 3.1,3.3] — Consider the plane, P , described by theequation(s) ?
(a) (10 pts) Write the system of equations in matrix-vector form(A?x = ?b).
(b) (10 pts) Find all solutions to the system of equations.x1+x3+x4 = 0 x1+x2+3x3+5x4 = 0
(c) (10 pts) Identify a basis for P .
(d) Next, consider the orthogonal projection onto P, defined byT(?x) = M?x, M ? R4×4. { We do not have the tools to derive thematrix M yet. }
i. (10 pts) What is the image of T [im(T ) = im(M )]?
ii. (10 pts) What is the dimension of the image of T [dim(im(T))]?
iii. (10 pts) What is the dimension of the kernel of T[dim(ker(T ))]?
iv. (10 pts) Identify a basis for ker(T )?
