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December 1st, 2018, 08:05 AM   #1
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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+b-c+d=0; c-2d=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?
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December 1st, 2018, 10:58 AM   #2
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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.
Thanks from Erke

Last edited by romsek; December 1st, 2018 at 11:52 AM.
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