prac03 - 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 3 (week 3) To get the most benefit from your practice class you should attempt all of these questions before the class. 1. Determine the null space and the column space of the matrix A = p 1 2 - 2 0 3 4 3 0 6 P geometrically. 2. Determine the null space and the column space of the matrix B = p 1 2 - 3 2 4 - 6 3 6 - 9 P geometrically. 3. Let X = bp 3 0 0 P , p 1 2 0 P , p 4 1 2 PB . a ) Show that X is a basis of R 3 . b ) Write the vector p 1 2 3 P as a linear combination of the vectors in X . 4. In each of the following, determine whether or not X is a basis of R 3 . a ) X = bp 1 0 3 P , p 2 1 - 1 P , p 1 - 5 4 PB . b ) X = bp 1 0 3 P , p 2 1 - 1 P , p - 1 - 2 11 PB . c ) X = bp 1 0 3 P , p 2 1 - 1 P , p 1 - 5 4 P , p 1 2 3 PB . 5. Let f 1 ( x ) = cos x , f 2 ( x ) = cos( x + 1) and f 3 ( x ) = sin
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