V is a subspace of R^4(adsbygoogle = window.adsbygoogle || []).push({});

V={(x, -y, 2x+y, x-2y): x,y E R}

1) extend {(2,-1,5,0)} to a basis of V.

2) find subspace W of R^4 for which R^4= direct sum V(+)W.

solution:

1)the dimension of V is 2.therefore i need to add one more vector to (2,-1,5,0).

the 2nd vector is (1,0,2,1).

therefore the basis is {(2,-1,5,0),(1,0,2,1)}.

i want to know whether my answer is correct.

2)dim of W is 2.

so i've to extend the basis for V by just adding any two vectors in R4, making sure that they don't become linearly dependent.

in this case i'm not able to find the basis.should i take the standard basis i.e. (1,0,0,0) or (0,1,0,0) or(0,0,1,0) as the first vector.

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# Vector space

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