Matrix Theory Midterm Examination II

Fall 1990

Instructions: There are five (5) questions on this examination, worth a total of 100 points. Work all problems.

1. (20 points) Solve the system:

\begin{aligned}c_{0} & + & c_{1} & + & c_{2} & + & c_{3} & = & 0 \\ c_{0} & + & i c_{1} & - & c_{2} & - & i c_{3} & = & 1 \\ c_{0} & - & c_{1} & + & c_{2} & - & c_{3} & = & 2 \\ c_{0} & - & i c_{1} & - & c_{2} & + & ic_{3} & = & 4 . \end{aligned} Give the inverse of the coefficient

matrix W.

2. (a) (10 pts) If P is the matrix that projects R^{n} onto a subspace S, explain why every vector in S is an eigenvector and so is every vector in S^{⊥}. What are the eigenvalues?

(b) (10 pts) Find the eigenvalues and eigenvectors and the diagonalizing matrix S for

A = [ \begin{matrix}1 & 0 \\ 2 & 3 \end{matrix} ] and B = [ \begin{matrix}7 & 2 \\ −15 & −4 \end{matrix} ]

3. (20 points) The following system has no solution:

Ax = [ \begin{matrix}1 & −1 \\ 1 & 0 \\ 1 & 1 \end{matrix} ] [ \begin{matrix}C \\ D \end{matrix} ] = [ \begin{matrix}4 \\ 5 \\ 9 \end{matrix} ] = b.

Sketch and solve a straight line fit that leads to the minimization of the quadratic (C − D − 4)^{2} + (C − 5)^{2} + (C + D − 9)^{2}. What is the projection of b onto the column space of A?

4. (a) (7 points) Find an orthonormal basis for the column space of

A = [ \begin{matrix}1 & 3 & 4 & 5 & 7 \\ −6 & 6 & 8 & 0 & 8 \end{matrix} ]^{T} .

(b) (7 points) Write A as QR, where Q has orthonormal columns and R is upper triangular.

(c) (6 points) Find the least squares solution to Ax = b, if b = (−3, 7, 1, 0, 4)^{T}.

5. (a) (10 points) If A is nonsingular, show that there must be some permutation P for which P A has no zeroes on its main diagonal.

(b) (10 points) Explain why the point (x, y) is on the line through (2, 8) and (4, 7) if

det [ \begin{matrix}x & y & 1 \\ 2 & 8 & 1 \\ 4 & 7 & 1 \end{matrix} ] = 0, or x + 2y = 18.