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December 1st, 2018, 07:05 AM  #1 
Newbie Joined: Dec 2018 From: kazakhstan Posts: 1 Thanks: 0  Linear algebra, sum of subspaces
Let U1, U2, U3 be subspaces of R^4: U1 = {(a,b,c,d):a=b=c} U2 = {(a,b,c,d):a+bc+d=0; c2d=0} U3={(a,b,c,d):3a+d=0} show that: a) U1 + U2 = R^4; b) U2 + U3 = R^4; c) U1 + U3 = R^4; whish of the sums are direct sum? 
December 1st, 2018, 09:58 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,404 Thanks: 1306 
can you write down the basis vectors for each of of the U's? If so for each problem use those to find the basis of the sum and use Gaussian elimination to find the resulting basis. If the equality is true you should end up with a 4x4 Identity matrix. U1 spans $R^2$ U2 spans $R^2$ U3 spans $R^3$ So if I understand the term direct sum it should be pretty obvious which pair of these is the only possible candidate for a direct sum. Last edited by romsek; December 1st, 2018 at 10:52 AM. 

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