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Spaces of Polynomials
4
...
and sets of matrices
...
q(x) =bo + b1x+···+ bnr, and t E JR?
...
x'1
1
Moreover, two polynomials p and q are equal if and only if ai =bi for 0 :: i :: n
...
(a) (2 + 3x + 4x2 + x3) + (5 + x - 2x2 + 7x3)
Solution:
(2+3x+4x2+x3)+(5+x-2x2+7x3) =2+5+(3+l)x+(4-2)x2+(1+7)x3
=7 + 4x + 2x2 + 8x3
(b) (3 + x2 - 5x3) - (1 - x - 2x2)
Solution:
(3+x2-5x3)-(l -x-2x2)
=
=
3-1 + [0-(-l)]x+ [1-(-2)]x2+ [-5-0]x3
2 + x + 3x2 - 5x3
EXAMPLE 1
(c)
5(2 + 3x
+
4x2 + x3)
(continued)
Solution: 5(2+3x+4x2+
...
x3=10+15x+20x2+5
...
x3)
Solution: 2(1 + 3x - x3) + 3(4 + x2 + 2
...
x3
+ 3(4) + 3(0)x + 3(1)x2 + 3(2)
...
x3
=14 + 6x + 3x2 + 4
...
These properties follow easily from the definitions of addition and scalar mul
tiplication and are very similar to those for vectors in !Rn
...
2
...
11 (Theorem 1
...
1) and of matrices
(Theorem 3
...
1)
...
When we look at polynomials in this way, it is the coefficients of the polynomi
als that are important
...
11 and matrices, we can also consider linear combinations of
polynomials
...
Definition
Let 13= (p1 (x),
...
Then the span
Span
of 13 is defined as
{p1 (x),
...
{1+x,1 +x3,x+x2,x+x3}
...
In particular, we have t1
EXAMPLE3
Determine if the set 13
-2, t2
=
=
=
3, t
3
=
3, and t4
=
1
...
Solution: Consider
0
=
=
t1 (1 +2x+2x2 - x3)+t2(3+2x+x2+x3)+t (2x2+2x3)
3
(t1 +3t2)+(2t1 +2t2)X+(2t1 +t2+2t )x2+(-t1 +t2+2t )X3
3
3
Comparing coefficients of the powers of x, we get a homogeneous system of linear
equations
...
Hence 13 is linearly independent
...
Is p(x)
1+5x - 5x2+x3 in the span
EXERCISE 1
Determine if :B
==
==
of :B?
EXERCISE 2
{1,x,x2,x3)
...
Consider :B
==
PROBLEMS 4
...
(2 - 2x+3x2+4x3) +(-3 - 4x+x2+2x3)
(b) (-3)(1- 2x+2x2+x3 + 4x4)
(c) (2 +3x+x2 - 2x3) - 3(1 - 2x+4x2 + 5x3)
(d) (2 +3x + 4x2) - (5 +x - 2x2)
(e) -2(-5 +x+x2) +3(-1 - x2)
( f) 2 ( � - tx+2x2) + H3 - 2x+x2)
(g) V2o +x+x2) + 7r(-1+x2)
Let :B
{1+x2+x3,2 +x+x3, -1+x+2x2+x3)
...
If a set is linearly dependent, find all
linear combinations of the polynomials that equal
the zero polynomial
...
(a) 0
2 +4x + 3x2+4x3
( c) -x+2x2+x3
(d) -4 - x+3x2
{1+2x+x2- x3,5x+x2,l- 3x+2x2+x3}
(b) {1+x+x2,x,x2+x3,3 +2x+2x2 - x3}
(c) {3 +x+x2,4 +x - x2,1+2x+x2+2x3,
-1+5x2+x3}
(d) {1+x+x3+x4,2 +x - x2+x3+x4,
x+x2+x3+x4}
Prove that the set :B
{1,x - 1,(x - 1)2} is linearly
(a)
==
(b)
-1+7x+Sx2+4x3
2 +x+Sx3
A4
==
independent and show that Span :B is the set of all
polynomials of degree less than or equal to
2
...
(3 +4x - 2x2+5x3) - (1 - 2x+Sx3)
(-2)(2 +x+x2+3x3 - x4)
(c) (-1)(2 +x+4x2+2x3) - 2(-1 - 2x - 2x2 - x3)
(d) 3(1+x+x3) +2(x - x2+x3)
(e) 0(1 +3x3 - 4x4)
(f) H3 - �x+x2) + !(2 +4x+x2)
(g) (1 + -Y2) (1 - -Y2+ < -Y2 - l ) x2 ) - H-2 + 2x2)
Let :B
{l+ x,x + x2,1 - x3}
...
If a set is linearly dependent, find all
linear combinations of the polynomials that equal
the zero polynomial
...
(a)
(b)
(c)
(d)
p(x)
p(x)
q(x)
q(x)
==
==
==
==
1
Sx+2x2+3x3
3 +x2 - 4x3
1+x3
{x2,x3,x2+x3+x4}
�
(b) {1+2•1
- �2• x+ �
6•x - �}
6
(c) {1 + x+x3,x+x3+ x5, 1 - x5 }
(d) (1 - 2x+x4,x - 2x2+ x5, 1 - 3x+x3}
(e) {l+2x+x2-x3, 2+3x-x2+x3+x4,1+x-2x2
+2x3+x4,1+2x+x2+x3 - 3x4,
4 + 6x - 2x2+Sx4)
Prove that the set 13
{1,x - 2,(x - 2 ) 2,( x - 2 ) 3}
(a)
B4
=
is linearly independent and show that Span :B is
the set of all polynomials of degree less than or
equal to
3
...
, Pk(x)} be a set of polynomials
of degree at most n
...
Span 13
...