Linear algebra, sum of subspaces

Dec 2018
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0
kazakhstan
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?
 

romsek

Math Team
Sep 2015
2,969
1,676
USA
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.
 
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