Name:
CSU ID:
Homework 6
October 9, 2015
1. Given the four vectors in <3 , which combinations are linearly independent?




~v1 = 
2
−5
4
3







,

~v2 = 
2
−1
2
7




,




~v3 = 


3
−1
4
5


,




~v4 = 
3
−6
4
12





2. Referring to the previous problem, what is the dimension of the space
spanned by the linearly independent vectors?
3. Referring to the vectors in the previous problem, which of the following
vectors are in the span(~v1 , ~v2 , ~v3 , ~v4 ).




(a) 
14
−17
22
26






(b) 



14
−18
22
26





4. In each part, let LA : <3 → <3 be multiplication by A, and let ~u1 =
[1, 0, 0]T , ~u2 = [2, −1, 1]T , and ~u3 = [0, 1, 1]T . Determine whether the
set {LA (~u1 ), LA (~u2 ), LA (~u3 )} is linearly independent in <3 .




1 1
1


(b) A =  1 1 −3 
2 2
0
1 1
2


(a) A =  1 0 −3  ,
2 2
0
5. Given that for the particular vectors ~v1 = [2, 3, −1]T , ~v2 = [3, 0, 7]T , ~v3 =
[5, −1, 6]T the linear operator satisfies


−1


L(~v1 ) =  2  ,
1
find




3


L(~v2 ) =  0  ,
7

7


(a) L  −2 
1

4


L(~v3 ) =  2 
5


0


(b) L  3 
1
Recall that a linear operator satisfies L(a~u + b~v ) = aL(~u) + bL(~v ).
Therefore, if ~v = a~v1 + b~v2 + c~v3 then L(v) can be determined.
6. Prove that if {~v1 , ~v2 , ~v3 } is a linearly independent set of vectors, then
so are {~v1 , ~v2 }, {~v1 , ~v3 }, and {~v2 , ~v3 }
7. (a) Show that the polynomials p1 (x) = 2 − 5x + 4x2 + x3 , p2 (x) =
1 + x − 6x2 + 7x3 , p3 (x) = −1 + 5x2 + 2x3 , p4 (x) = −3 − 14x +
27x2 − 28x3 are linearly independent.
(b) Express the polynomial p(x) = 1 + x + x2 + x3 as a linear combination of the pi , i = 1 : 4.
(c) Express the polynomial q(x) = 5 − x + 3x3 − 7x3 as a linear
combination of the pi , i = 1 : 4.
8. Show that the set of polynomials of degree less than or equal to 3, P 3 ,
such that p(7) = 0 is a subspace of P 3 . What is it’s dimension?