# Therefore x is a linearly independent subset of r4

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: namely a = b = c = 0. Therefore X is a linearly independent subset of R4 . Note that any linear combination of the vectors in X will result in a vector whose ﬁnal entry is zero. Hence X does not span R4 . (Also, dim(R4 ) = 4, and we would need 4 linearly independent vectors to span R4 .) The dimension of Span(X ) is 3. Math 2061: Tutorial 4 (week 5) — Solutions Page 2 Linear Mathematics 1 1 1 5. Let X = Tutorial 4 (week 5) — Solutions 1 1 0 , 1 0 0 , 3 2 0 , Page 3 . 3 a) Show that X spans R . b) Explain why X is not a basis for R3 . c) Find a subset of X which is a basis for R3 . Solution a) X spans R3 if there is a solution, for every x y z 1 1 1 =a +b 1 1 0 +c x y z 1 0 0 ∈ R3 , to the equation +d 3 2 0 1113 1102 1000 = a b c d . The augmented matrix for this system of equations, and a row echelon form is 1113x 1102y 1000z 1113 x 0 1 1 3 x−z 0 0 1 1 x−y Row reduce −−− −−→ So there are inﬁnitely many solutions for any x y z . ∈ R3 , and hence X spans R3 . b) Any basis of R3 contains exactly 3 vectors, and X contains 4 vectors, so it is not a basis. c) Looking at the row echelon form of the matrix in part a), we see that column 4 of the coefﬁcient matrix does not contain a leading one. Removing column 4 from the coefﬁ111 cient matrix gives us the matrix 1 1 0 , which would reduce to I3 . Hence the subset 100 1 1 1 , 1 1 0 , 1 0 0 of X is linearly independent, and is a basis for R3 . 6. You are given the following data points: x0 2 4 y 7 21 43 Construct a Lagrange basis {p0 , p1 , p2 } of P2 using the x values from the data set. Hence ﬁnd the unique quadratic p that ﬁts the data exactly. Estimate the value of y when x = 1. Solution By deﬁnition, the vectors in the Lagrange basis of P2 are as follows: 1 (x − 2)(x − 4) = (x − 2)(x − 4), (0 − 2)(0 − 4) 8 (x − 0)(x − 4) 1 p 1 ( x) = =...
View Full Document

## 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.

Ask a homework question - tutors are online