Math 333 First Exam (9 Oct 2014)
Name:
If you use a theorem indicate what it is.
instructed otherwise).
Good Luck!
Show all work (unless
0. Circle True or False without explanation:
( T or F ) Every non-zero vector v in V belongs to some basis of V .
( T or F ) The spaces P3 and M2×2 are isomorphic (to each other).
( T or F ) If {v1 , . . . , vn } is independent then so is {T v1 , . . . , T vn } (for a linear T ).
( T or F ) If A is a square matrix and A2 = 0 then A = 0.
( T or F ) For any b in Z8 , the equation 6x = b has a solution x in Z8 .
1. Find a basis and the dimension for the space A3×3 of all 3 × 3 antisymmetric
matrices. Start by filling-in the six blanks in the general from of such a matrix.
A3×3


:= 

a


b


c : a, b, c ∈ R . Do not prove anything.

2. Quickly and decisively show that the set In×n of all invertible n × n matrices is
not a linear subspace of Mn×n (for any natural n).
3. Carefully prove that the following subset W of F is a subspace:
W := {f ∈ F : f (2x) = f (x) for all x ∈ R},
(Plainly, W consists of all functions on R unchanged by the substitution x 7→ 2x.)
4. Give a bullet-proof verification of independence of {1, ex , e−x } in F .
(If too hard, do {ex , e−x } for partial credit.)
5. Consider two transformations T and S on Mn×n given by
1
S(A) := (A + AT )
2
1
and T (A) := (A − AT ).
2
a) Carefully verify that S is linear.
b) Compute and simplify the formula for the composition S ◦ T
S ◦ T (A) = . . .
6. Consider T : P2 → P2 given by T (p(x)) := p(x) + p′ (x).
a) Compute the matrix [T ]B←B when B = {1, x + 1, x2 }.
b) Determine the kernel of T .
c) Based on b), can you tell readily if T is onto. Explain.
7. Prove that if T : V → W is linear and ker(T ) = {0} then T is one-to-one.
8. A binary code encodes x = (x1 , x2 , x3 )T to (b1 , b2 , b3 , b4 , b5 )T := Gx where G is
5 × 3 matrix of rank three and with rows satisfying
r1 + r2 + r3 = r4 + r5
and r2 = r4 + r3 .
Find the conditions for (b1 , b2 , b3 , b4 , b5 )T to be an uncorrupted message (i.e., b = Gx
for some x). Explain your reasoning!