Name:
CSU ID:
Homework 6
March 6, 2015
1. Given the four vectors in <3 , which combinations are linearly independent?


2


~v1 =  −5  ,
4


2


~v2 =  1  ,
−6



−1


~v3 =  0  ,
5

2


~v4 =  4 
−3
2. S4.2 ]12(a),(b)
3. S4.2 ]20
4. Consider the polynomials p1 (x) = 2 − 6x + 4x2 , p2 (x) = −1 + x + x3 ,
p3 (x) = 2 + 3x3 .
(a) Is the polynomial p(x) = 1 + 7x − 4x2 + 7x3 dependent or independent of those given? Do the polynomials p1 , p2 , p3 , p span the
vector space of polynomials of degree less than or equal to 3.
(b) Is the polynomial q(x) = 2 + 7x − 4x2 + 7x3 dependent or independent of those given? Do the polynomials p1 , p2 , p3 , q span the
vector space of polynomials of degree less than or equal to 3.
5. Which of the following sets of polynomials are linearly independent in
P 3?
(a) {2−5x+4x2 +x3 , 2+x−6x2 +7x3 , −1+5x2 +2x3 , 2+4x−3x2 +x3 }
(b) {2 − 5x + 4x2 + x3 , 2 + x − 6x2 + 7x3 , −1 + 5x2 + 2x3 , −3 − 14x +
27x2 − 28x3 }
(c) {2 + x − 6x2 + 7x3 , −1 + 5x2 + 2x3 , 2 + 4x − 3x2 + x3 , 7 + 9x −
17x2 + 7x3 }
6. Referring to the previous problem, which sets of polynomials form a
bases for P 3 , polynomials of degree less than or equal to 3.
7. Consider the following matrices
"
A=
2 −5
4
1
#
"
,
B=
2 1
−6 7
#
"
,
C=
−1 0
5 2
#
"
D=
2 4
−3 1
#
"
E=
−3 −14
27 −28
#
"
F =
7 9
−17 7
#
Referring to the previous problem,
(a) Which matrices correspond to the polynomials in 5(a)? Do they
form a basis for 2 × 2 matrices?
(b) Which matrices correspond to the polynomials in 5(b)? Do they
form a basis for 2 × 2 matrices?
(c) Which matrices correspond to the polynomials in 5(c)? Do they
form a basis for 2 × 2 matrices?
8. S4.3 ]16(a),(b)
9. 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?