Linear Algebra 2270
Homework 5
preparation for the quiz on 06/24/2015
Problems:
1. Let W1 be a subspace of R4 defined as follows:
 
x1
x2 
4

W1 = {
x3  ∈ R : x1 , x2 , x3 , x4 ∈ R, x1 + x2 − 2x3 − x4 = 0, x1 + 2x2 + 2x3 = 0}
x4
Consider an arbitrary W , which is a subspace of W1 (W ≺ W1 ). What is the biggest dimension
that W can have? Explain.
2. Consider a vector v1 given below. Enrich the set {v1 } to a basis of R2 . In other words find v2 such
that {v1 , v2 } is a basis of R2 .
1
v1 =
−1
3. Consider independent vectors v1 , v2 given below. Enrich the set {v1 , v2 } to a basis of R4 . In other
words find v3 , v4 such that {v1 , v2 , v3 , v4 } is a basis of R4 .
 
 
1
1
1
1

 
v1 = 
0 , v2 = 1
0
1
4. Consider the vector space W1 defined in problem 1. Consider a v1 ∈ W1 given below. Enrich the
set {v1 } to a basis of W1 .
 
−4
1

v1 = 
1
−5
5. Consider a space
V = span(f1 , f2 , f3 )
where f1 , f2 , f3 ∈ RR are such that for any x ∈ R:
f1 (x) = x − 1, f2 (x) = sin(x), f3 (x) = cos(x)
Let v1 ∈ V be a function such that for any x ∈ R:
v1 (x) = sin(x) + 3 cos(x) + 4 − 4x
Enrich the set {v1 } to a basis of V .
6. Determine if the following functions are 1-1 and onto. If a function f is both 1-1 and onto, find its
inverse f −1 .
(Hint: to find an inverse of a function f , one can consider an equation f (x) = y and “solve for x”,
thus arriving at x = f −1 (y))
(a) f : R → R, f (x) = x
1
(b) f : R → R, f (x) = x3 − 2
(c) f : R → R, f (x) =
1
x2 +1
(d) f : R → [−1, 1], f (x) = sin(x)
(e) f : [0, ∞) → R, f (x) = x2
x1
2
(f) f : R → R, f
= x1 + x32
x2
x1
x1
2
2
(g) f : R → R , f
=
x2
x1 + 4x2
2