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I am given two sub-spaces, v1 and v2. They are in the vector space $\mathbb{R}[x]_{\deg < 4}$.

$v_1=\text{span} \left( {x}^{3}+4\,{x}^{2}-x-3,{x}^{3}+5\,{x}^{2}+5 ,3\,{x}^{3}+10\,{x}^{2}-5\,x+3\right) $ $v_2=\text{span} \left( {x}^{3}-5\,x,{x}^{2}+x,{x}^{3}+2\,{x}^{2}-3 \,x\right)$

I am told that one sub-space is included in the other, and I need to

a. determine which subspace is included in the other

b. find the base of the smaller subspace

c. complete the base from the part b of the question so that it is the base of the larger subspace.

So far I've understood that $v_2$ is part of $v_1$ because $v_1$ has scalars without x-dependence, and $v_2$ does not have any. So $v_1$ includes $v_2$. Next for b I rref-ed the matrix of $v_2$ and found that the 3 vectors are linearly independent, and since I am told they are span therefore I know it is the base. Found. For c I need to add something so that it is the base of $v_1$. This is where I'm not certain how to proceed.

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    yes you're right, this is my mistake.2011-12-27

1 Answers 1

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wrt the basis $x^3,x^2,x,1$, the spaces are $V_1=\langle(1,4,-1,-3),(1,5,0,5),(3,10,-5,3)\rangle$ $ V_2=\langle(1,0,-5,0),(0,1,1,0),(1,2,-3,0)\rangle $ with a little manipulation (rref on the appropriate matrices) you get $V_1=\langle(1,0,-5,0),(0,1,1,0),(0,0,0,1)\rangle$ $ V_2=\langle(1,0,-5,0),(0,1,1,0)\rangle $ so you can see that $V_2\subset V_1$ and that $V_1=V_2+\langle(0,0,0,1)\rangle$