Math 316
Carter
Test 1 Spring 2011
General Instructions: Do not write on this test. Write your name on only the outside
of your blue books. Do all your work inside your blue books. On the inside of your blue book,
write neat complete answers to the questions below. Show your work. It is not considered
appropriate to chant, “Bring on the show! Bring on the show!” at a chamber music concert.
Good luck.
1. (20 points) Let X = [1, 1, 0, 2]t and Y = [1, 1, 1, −3]t determine |X|, |Y |, X · Y , and
the cosine of the angle between the vectors.
2. (20 points) Let A ∈ M (n, n) be a square matrix the columns of which form an orthonormal basis for Rn = M (n, 1). Determine the inverse of A and justify that this is
the inverse. [Hint: this requires little to no direct calculation].
3. (20 points) Use the Gram-Schmidt process to find an orthogonal basis for the solution
set to the equation
x + y + z + w = 0.
4. (20 points) Use the calculation of the previous problem to factor the matrix


−1 −1 −1 1


 1
0
0 1 


A=
1
0 1 

 0
0
0
1 1
as A = QDU where the columns of the matrix Q are an orthonormal basis for M (4, 1),
the matrix D is diagonal, and the matrix U is unipotent.
5. (20 points) In class, I have said that the inner product between two vectors, X, Y ∈
M (n, 1) induces a positive definite, symmetric, bilinear form. This means that:
(a) (positive definite) X · X ≥ 0 and X · X = 0 if and only if X = 0;
(b) (symmetric) X · Y = Y · X;
(c) (bilinear)
(αA + βB) · C = α(A · C) + β(B · C),
and
A · (αB + βC) = α(A · B) + β(A · C).
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Moreover, if X = [x1 , x2 , . . . , xn ]t and Y = [y1 , y2 . . . , yn ]t , then X · Y =
Pn
j=1
xj y j .
Prove that
(αX1 + βX2 ) · Y = α(X1 · Y ) + β(X2 · Y ).
If you have time (after you have completed the rest of the test), prove the other
properties.
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