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  3. 5. [§ 3.1,3.3] — consider the plane, p , descri

5. [§ 3.1,3.3] — Consider the plane, P , described by the equation(s) ? (a) (10 pts)...

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Advance Math

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 )?

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