Could someone please tell me how to approach this?
Determine the dimension of the smallest subspace of $\Bbb R^4$ that contains vectors $(0, 1, 0, 1), (3, 4, 1, 2), (6, 4, 2, 0)$ and $(−3, 1, −1, 3)$.
Determine the dimension of the smallest subspace of $\Bbb R^4$ that contains vectors $(0, 1, 0, 1), (3, 4, 1, 2), (6, 4, 2, 0)$ and $(−3, 1, −1, 3)$.
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linear-algebra
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5Put the vectors into the rows of a matrix and perform Gaussian elimination on the rows. Next question? – 2017-01-27
1 Answers
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Let $v_1=(0,1,0,1), v_2=(3,4,1,2),v_3=(6,4,2,0),v_4=(-3,1,-1,3).$
We have $v_2+v_4=(0,5,0,5)=5v_1.$ So $v_4=(1/5)v_1-v_2.$
We have $v_3-2v_2=(0,-4,0,-4)=-4v_1.$ So $ v_3=2v_2-4v_1.$
Therefore $v_3$ and $v_4$ belong to any linear subspace that contains $v_1$ and $v_2.$
Verify that $v_1,v_2$ are linearly independent. So the subspace in question is $\{av_1+bv_2: a,b\in \mathbb R\}$ which is $2$-dimensional.