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Ma1bAnHw2Sol

# Ma1bAnHw2Sol - CALIFORNIA INSTITUTE OF TECHNOLOGY...

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CALIFORNIA INSTITUTE OF TECHNOLOGY Department of Mathematics Math 1b; Solutions to Homework Set 2 Due: January 22, 2008 1. We will see later in the course that the solution space of m independent homogenous linear equations in n m variables is of dimension n - m ; for the moment we prove this by ad hoc methods in the special cases in this problem. In HW1 we saw that in (1)-(4), S is the subspace defined by one homogeneous linear equation, so we expect dim( S ) to be 3 - 1 = 2. We calculate that the following pairs of vectors are solutions to the respective equations: (1) e 2 , e 3 ; (2) e 1 - e 2 , e 3 ; (3) e 1 - e 2 , e 1 - e 3 ; (4) e 1 + e 2 , e 3 . Let P be the set consisting of the pair of vectors; visibly P is independent, so P is a basis for L ( P ) and hence dim( L ( P )) = 2. Thus it remains to show that S = L ( P ). As P S , L ( P ) S by Problem 2 on this HW. Thus 2 = dim( L ( P )) dim( S ) dim( V 3 ) = 3 , by Problem 3 on this HW, so dim( S ) = 2 or 3. Further in the first case dim( S ) = dim( P ) so P = S by part (b) of Problem 3, so we may assume dim( S ) = 3. Then S = V by

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