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Unformatted text preview: he null space of A, so nullity(A) = 2.
Hence, rank(A) + nullity(A) = 3 + 2 = 5, which is the number of columns of A.
8. Find a basis for the subspace V =
Solution A vector w
x
y
z w
x
y
z ∈ R4 2x − y = z − 3w of R4 . ∈ R4 belongs to V if and only if 2x − y = z − 3w or, equivalently, if y = 2x − z + 3w. Thus, an arbitrary element
w
x
y
z Math 2061: Tutorial 4 (week 5) — Solutions = w
x
2x−z +3w
z =w w
x
y
z of V can be written in the form 1
0
3
0 +x 0
1
2
0 +z 0
0
−1
1 . Page 4 Linear Mathematics Tutorial 4 (week 5) — Solutions
1
0
3
0
0
0
−1
1 Therefore, the set
a 1
0
3
0 +b 0
1
2
0 +c ,
= 0
1
2
0 ,
0
0
0
0 0
0
−1
1 Page 5 spans V . It is also linearly independent, since if , we can see immediately that a = b = c = 0.
1
0
3
0 Therefore, V is a subspace of R4 with basis 0
1
2
0 , , 0
0
−1
1 . 9. Let X = {p1 , p2 , p3 } be a subset of P2 , where p1 (x) = x2 − x + 2, p2 (x) = 3x + 4 and
p3 (x) = 2x2 + x + 8. Does Span(X ) equal P2 ? Find the dimension of Span(X ).
Solution Let f be an arbitrary polynomial in P2 deﬁned by the formula f (x) = αx2 + β x2 + γ
where α, β , γ ∈ R. Then f belongs to Span(X ) if and only if there exist a, b, c ∈ R such that
ap1 (x) + bp2 (x) + cp3 (x) = αx2 + β x + γ ; that is,
a(x2 − x + 2) + b(3x + 4) + c(2x2 + x + 8) = αx2 + β x + γ .
Equating coefﬁcients of x2 , x and x0 , we have
a
+ 2c = α
− a + 3b + c = β
2a + 4b + 8c = γ
The augmented matrix for this system of equations is
102α
−1 3 1 β
248γ
and this reduces to 102
011
000 , α
β +α
3
3γ −4β −10α
3 . This means that there are values of α, β , γ for which the equations are inconsistent, and hence
there are polynomials in P2 that cannot be written as linear combinations of p1 , p2 , p3 . (Note
that it also means that p1 , p2 , p3 are linearly dependent, since the equation
a(x2 − x + 2) + b(3x + 4) + c(2x2 + x + 8) = 0
would have nonzero solutions for a, b, c.)
So Span(X ) is not equal to P2 , and its dimension is less than 3...
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This note was uploaded on 06/04/2013 for the course MATH 1002 taught by Professor Cartwright during the One '08 term at University of Sydney.
 One '08
 Cartwright
 Linear Algebra, Algebra, Vector Space

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