Mathematics 2270
PRACTICE EXAM II
Fall 2004
1. Exercises 4.1.13, 4.1.14, 4.1.21 4.1.48, 4.2.12 4.2.19 4.2.56 4.2.60, 4.3.12 4.3.16 4.3.38
4.3.39, 5.1.26
2. We note F (R, R) the vector space of real functions (with the usual addition and scaling
of functions).
a) Prove that P = {f ∈ F (R, R) | ∀x ∈ R, f (−x) = f (x)} is a vector subspace of
F (R, R).
b) Prove that the following map is linear
T : F (R, R) → F (R, R)
f 7→ (x 7→ f (x) + f (−x))
Prove that Im(T ) = P .
3. Let
 

1
1
1 
0
4
 

v1 = 
−1 , v2 0 , V = span {v1 , v2 } ⊂ R
0
0

a) Find an orthonormal basis (u1 , u2 ) of V .
b) Write down the standard matrix of projV .
c) Let
 
 
0
1


1 
0
−1

u3 = √ 
, u4 = 



0
1
3
1
0
Prove that B = (u1 , u2 , u3 , u4 ) is a basis of R4 . Then compute the B-matrix of projV .
4. a) Prove that complex conjugation is a linear map form C to C, we will denote it J.
b) Prove that (1, i) is a basis of C. Find the matrix of J in this basis.
c) Prove that the following map is an isomorphism, and write down its inverse:
T : C → R2
R(z)
z 7→
I(z)
d) The map T ◦ J ◦ T −1 goes from R2 to R2 . Write down its standard matrix. Prove
then that it is equal to a reflection.
5. (Direct sums, complements) Let V be a vector space, and dim V = n. Let W , U be
two vector subspaces of V
A) Assume that W ∩ U = {0} and
dim W + dim U = dim V
a) Let u1 , . . . , ul ∈ U , w1 , . . . , wk ∈ W two sets of linearly independent vectors. Prove
that u1 , . . . , ul , w1 , . . . , wk are linearly independent. Deduce that if u1 , . . . , um ∈ U is
a basis of U and w1 , . . . , wp ∈ W is a basis of W , then u1 , . . . , um , w1 , . . . , wp is a basis
of V .
b) Prove then that for all x ∈ V there exists a unique pair of vectors u ∈ U w ∈ W
such that x = u + w.
B) Now assume that for all x ∈ V there exists a unique pair of vectors u ∈ U and
w ∈ W such that x = u + w. Prove that then W ∩ U = {0} and that
dim U + dim W = dim V
Remark: W and V are said to be complement one of each other. Equivalently you
can say that V = W ⊕ U (read ⊕ as “direct sum”).