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Math 221, Winter 2005, Term 2 Page 1 of 4
Final Exam
April 22, 2006
No books. No notes. No calculators. Time: 15:30 - 18:00, which is 150 minutes.
Problem 1. (8 points) Find the general solution in parametric vector form of the inhomogeneous system of linear equations
x 3 + x 4 = 7 x 2 + x 3 = 5 x 1 + x 2 = 3 x 1 + x 4 = 5
Problem 2. (8 points) Find the inverse of the matrix (^)
Problem 3. (10 points) Consider the matrix
A =
(a) Find a basis for the column space of A. (b) Find a basis for the nullspace of A.
Problem 4. (5 points) (a) What is the rank of a 4 × 5 matrix whose null space is three dimensional? (b) What is the rank of a 4 × 4 matrix with determinant −1?
Problem 5. (6 points)
The colour of light can be represented in a vector
R
G
B
, where R = amount of red,
G = amount of green and B = amount of blue. The human eye and the brain
transform the incoming signal into the signal
I
L
S
, where
I = intensity =
R + G + B
L = long-wave signal = R − G
S = short-wave signal = B −
R + G
(a) Find the matrix P representing the transformation from
R
G
B
(^) to
I
L
S
(b) Consider a pair of yellow sunglasses for water sports which cuts out all blue light and passes all read and green light. Find the matrix A which repre- sents the transformation incoming light undergoes as it passes through the sunglasses. (c) Find the matrix for the composite transformation which light undergoes as it first passes through the sunglasses and then the eye.
Problem 6. (6 points) Find the determinant of the matrix
1 + a 1 1 1 1 1 + b 1 1 1 1 1 + c 1
Problem 7. (10 points) Consider the system of differential equations
~x ′^ = A~x where A =
(a) Classify the origin as either an attractor, repellor, or saddle point. Justify your answer.
(a) Compute the matrix A of Q. (b) Perform an orthogonal change of variables ~x = P ~y, that transforms Q into a quadratic form without cross-product term. Give P and the new quadratic form. (c) Give a geometric description of the set of all vectors ~x for which Q(~x) = 0 (i.e. is it a point, a single line, a union of lines, a hyperbola, or an ellipse?). Justify your answer and draw a sketch.
Problem 13. (6 points) Let E be the plane in R^3 spanned by the orthogonal vectors
~v 1 =
(^) and ~v 2 =
The reflection across E is the linear transformation R : R^3 → R^3 defined by the formula R(~x) = 2 projE (~x) − ~x (a) Compute R(~x) for
~x =
(b) Find the eigenspace of R corresponding to the eigenvalue 1. That is, find the set of all vectors ~x for which R(~x) = ~x. Justify your answer.
Problem 14. (6 points) Suppose E is the column space of the matrix
A =
Then E is a plane in R^3. Determine the orthogonal projection of
~b =
onto E. (Hint: The columns of A are not orthogonal! Use the fact that if ˆx is any
least square solution of A~x = ~b, then Axˆ = projE (~b ).)
Problem 15. (4 points) Suppose A is a 3 × 3 matrix such that A^3 = 0. (a) Explain why A cannot be invertible. (b) Show that (I − A + A^2 ) is invertible.
