8 find a basis for the subspace v solution a vector w

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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 non-zero 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.

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