tut01 - 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 Tutorial 1 1. a ) Calculate the matrix product 3 0 4 1 1 2 - 1 3 5 5 - 2 4 . b ) Hence find the solution to the system of linear equations that corresponds to the augmented matrix 3 0 4 31 1 1 2 11 - 1 3 5 9 . 2. Each of the following matrices is the reduced row echelon form of an augmented matrix be- longing to a system of linear equations in the variables x i , ( i = 1 , 2 ,... ) . (Both the systems represented here have infinitely many solutions – why?) For each augmented matrix (i) determine the number of parameters needed to solve the system and (ii) express the solution of the system in parametric form. a ) parenleftBigg 1 0 4 0 0 1 - 5 - 1 0 0 0 0 parenrightBigg b ) parenleftBigg 1 2 0 0 6 0 0 1 0 5 0 0 0 1 - 1 parenrightBigg 3. Recall that R 3 = braceleftBigparenleftBig x 1 x 2 x 3 parenrightBigvextendsingle vextendsingle vextendsingle x 1 ,x 2 ,x 3 R bracerightBig is a vector space. Describe each of the following
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