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prac04 - T HE U NIVERSITY OF S YDNEY P URE M ATHEMATICS...

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T HE U NIVERSITY OF S YDNEY P URE M ATHEMATICS Linear Mathematics 2012 Practice session 4 1. Find a basis of the subspace V = braceleftBigparenleftBig w x y z parenrightBig R 4 vextendsingle vextendsingle vextendsingle 2 x - y = z - 3 w bracerightBig of R 4 . 2. Let X = braceleftBigparenleftBig 1 1 1 parenrightBig , parenleftBig 1 1 0 parenrightBig , parenleftBig 1 0 0 parenrightBig , parenleftBig 3 2 0 parenrightBigbracerightBig . a ) Show that X spans R 3 . b ) Explain why X is not a basis for R 3 . c ) Find a subset of X which is a basis for R 3 . 3. You are given the following data points: x - 1 0 1 2 y 4 1 - 2 1 Construct a Lagrange basis { p 0 , p 1 , p 2 , p 3 } of P 3 using the x values from the data set. Hence find the unique cubic polynomial p ( x ) that fits the data exactly. Estimate the value of y when x = 1 2 . 4. The matrix J is a reduced row echelon form of the matrix A . Let A = parenleftbigg 1 1 3 1 0 - 1 0 1 4 0 0 0 parenrightbigg . a ) Find a reduced row echelon form, J , of the matrix A .
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