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Math 205A Final Exam (75 points)
Name:
- Check that you have 8 questions on four pages.
- Show all your work to receive full credit for a problem.
- (12 points) Short answers: (Show all the calculations to get the answers. No explanations needed.)
(a) If C is a 4 × 5 matrix, what is the largest possible rank of C? What is the smallest possible dimension of Nul C?
(b) For a 3 × 3 matrix B, det B = −1. Find det 4B.
(c) Find the distance between the vector ~u =
[
]
and the vector ~v =
[
]
(d) Let T : R^3 → R^2 be the linear transformation given by
T (x 1 , x 2 , x 3 ) = (x 1 − x 2 , 2 x 2 − x 3 ).
Find a matrix A such that T (~x) = A~x.
(e) Let ~p 1 (t) = 1, ~p 2 (t) = 2t, ~p 3 (t) = 4t^2 − 2 and ~p 4 (t) = 8t^3 − 12 t. Then B ={~p 1 , ~p 2 , ~p 3 , ~p 4 }
is a basis for P 3. Find the polynomial ~q in P 3 , given that [q]B =
- (4 points) Suppose the columns of a 4 × 4 matrix A span R^4. Is det A = 0? Explain.
(c) W = {all symmetric matrices in M 2 × 2 }. (Recall that a symmetric matrix is a matrix A such that AT^ = A.)
- (5 points) Suppose A is a symmetric n × n matrix and B is any n × n matrix. Explain why BABT^ is orthogonally diagonalizable.
- (12 points) Define a linear transformation T : P 1 → R 2 by T (~p) =
[
~p(1) ~p(1)
]
. (Recall that a
vector ~p in P 1 is a polynomial of the form a + bt.)
(a) Find T (3) and T (2 − 7 t).
(b) Find a polynomial p in P 1 such that T (~p) =
[
]
or explain why we cannot find such a polynomial.
(c) Find a polynomial that spans the kernel of T. (Recall that the kernel of T is the space of all vectors that are mapped to the zero vector under T ,i.e., the kernel is the null space of T .)
(d) Is T one-to-one? Explain.
- (10 points) Hurricanes develop low pressure at their centers that generates high winds. The maximum wind speed s (in knots) and the central pressure p of a hurricane are approximately related by the equation b 0 + b 1 p = s. We have the following data on four recent Atlantic hurricanes in the United States.
p 905 920 960 990 s 130 110 80 60
Find b 0 and b 1 so that the model b 0 + b 1 p = s is a least-squares fit to the data. (Start by using the given data to write a system of linear equations to determine b 0 and b 1 .)
- (8 points) Let ~u 1 =
(^) and ~u 2 =
. Let W = Span{~u 1 , ~u 2 } and ~y =
(a) The set {~u 1 , ~u 2 } is not an orthogonal set. Find two vectors in W that are orthogonal to each other and span W. (Use the vectors ~u 1 and ~u 2 to produce the two orthogonal vectors.)
(b) Find a vector in W that is closest to ~y.
