Name:
CSU ID:
Homework 5
October 2, 2015
Include this list of problems stapled to the homework turned in.
1. Show that the span of A1 , A2 , A3 and A4 is all 2 × 2 matrices.
"
A1
"
A3 =
#
−2 0
0 0
"
,
0 0
−4 0
A2 =
#
"
,
A4 =
#
0 3
0 0
0 0
0 6
#
2. Show that the span of p1 (x) = −2, p2 (x) = 3x, p3 (x) = −4x2 , and
p4 (x) = 6x3 is all polynomials of degree less than or equal to 4.
3. Show that the span of~v1 , ~v2 , ~v3 and ~v4 is <4 , real vectors of length 4.




~v1 = 
−2
0
0
0




,




~v2 = 
0
3
0
0




,




~v3 = 
0
0
−4
0



,





~v4 = 
0
0
0
6





4. Redo the previous problems given the matrices defined below.
"
A1 =
2 −1
4
6
#
"
,
A2 =
7 5
3 1
#
"
,
A3 =
11 4
2 3
#
"
,
A4 =
5. Let V be the vector
space all polynomials
of degree less than or equal
to 3. Let S = bx + cx3 ; a, b ∈ < . Either prove that S is a subspace
or show a counter example.
6. Let V be the vector space all polynomials
of degree less than or equal to
3. Let S = 1 + bx + cx3 ; a, b ∈ < . Either prove that S is a subspace
or show a counter example.
7. Show that the set of all points in <n lying in a hyperplane (c1 x1 +
c2 x2 + · · · + cn xn = d) is a vector space with respect to the standard
operations of vector addition and scalar multiplication if and only if
the plane passes through the origin. Hint: Consider what d must be.
Then write the equation in vector form.
−2
0
4 −1
#
8. Which of the following in (a)-(d) are linear combinations of
"
A=
2 0
1 4
#
"
,
"
(a)
"
(c)
B=
−18 31
−9 −3
−11
2
−15 −6
−3 5
−2 0
#
"
,
C=
#
"
(b)
#
"
(d)
3 30
10 21
1 6
3 5
#
#
−11
2
−14 −6
#
9. Given the polynomials
p1 (x) = 2 + x2 + 4x3
p3 (x) = 1 + 6x + 3x2 + 5x3
p2 (x) = −3 + 5x − 2x2
p4 (x) = 1 − x2 + x3
find a linear combination of them to reproduce the following polynomials.
(a) q1 (x) = 1 + x
(b) q2 (x) = x − x2 + 3x3
Is there any polynomial of degree less than or equal to 3 that cannot
be expressed as a linear combination of p1 , p2 , p3 , p4 . Why or why not?