Linear Algebra 2270
Homework 4
preparation for the quiz on 06/18/2015 (Thursday!!)
Problems:
1. A linear system is called homogeneous if the rhs is 0. Prove that a solution set of a homogeneous
linear system is a subspace of the dimension equal to the number of free variables.
Hint: Use the parametric representation of the solution set:
{u = q0 + a1 q1 + . . . + ak qk ∶ a1 , . . . , ak ∈ R}
where k is the number of free variables and q1 , . . . , qk are linearly independent (we proved that in
the lecture).
Additionally use the fact (you do not have to prove this fact) that if the system is homogeneous,
then the row operations on the augmented matrix will keep it homogeneous and thus q0 = 0.
2. Prove that v = {v1 , v2 } is a basis of R2 and find a representation of vector w in this basis, thus find
a
a, b ∈ R such that w = [ ] = av1 + bv2 , where
b v
2
−1
−2
v1 = [ ] , v 2 = [ ] , w = [ ]
0
1
4
3. Prove that v = {v1 , v2 , v3 } is a basis of R3 and find a representation of vector w in this basis, thus
⎡a⎤
⎢ ⎥
⎢ ⎥
find a, b, c ∈ R such that w = ⎢ b ⎥ = av1 + bv2 + cv3 , where
⎢ ⎥
⎢c⎥
⎣ ⎦v
⎡0⎤
⎡0⎤
⎡−1⎤
⎡2⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
v1 = ⎢0⎥ , v2 = ⎢ 1 ⎥ , v3 = ⎢1⎥ , w = ⎢4⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢4⎥
⎢2⎥
⎢0⎥
⎢0⎥
⎣ ⎦
⎣ ⎦
⎣ ⎦
⎣ ⎦
4. Find the dimension of the following subspaces W . (you need to find a basis of W or use problem 1)
0
(a) V = R2 , W = {[ ] ∈ R2 ∶ x ∈ R}
x
x
(b) V = R2 , W = {[ 1 ] ∈ R2 ∶ x1 , x2 ∈ R, x1 + x2 = 0}
x2
⎡x1 ⎤
⎢ ⎥
⎢ ⎥
3
(c) V = R , W = {⎢x2 ⎥ ∈ R3 ∶ x1 , x2 , x3 ∈ R, x1 + x2 − 2x3 = 0}
⎢ ⎥
⎢x3 ⎥
⎣ ⎦
⎡1⎤ ⎡ 1 ⎤ ⎡ 3 ⎤
⎛⎢⎢ ⎥⎥ ⎢⎢ ⎥⎥ ⎢⎢ ⎥⎥⎞
(d) W = span ⎜⎢2⎥ , ⎢ 0 ⎥ , ⎢ 2 ⎥⎟
⎝⎢⎢0⎥⎥ ⎢⎢−1⎥⎥ ⎢⎢−2⎥⎥⎠
⎣ ⎦ ⎣ ⎦ ⎣ ⎦
R
(e) V = R , W is a set of polynomials up to order 2. In other words
W = {f ∈ RR ∶ there are three numbers a0 , a1 , a2 ∈ R such that for all x ∈ R, f (x) = a0 +a1 x+a2 x2 }
5. (an extra problem, will not appear on the quiz) Consider the vector space of functions RR . Consider
a function equal to 1 at a particular point and 0 everywhere else:
fxo (x) = {
1 ∶ x = x0
0 ∶ x ≠ x0
Prove that if x1 , . . . , xn ∈ R are distinct points, then the vectors fx1 , . . . , fxn are linearly independent.
Use that fact to show that RR is not finite dimensional.
1