Finding the Basis of a Subspace $\mathbb{R}^3$ that is spanned by a series of Vectors

linear algebravector-spacesvectors

The Question goes as follows:

Find a basis for the subspace of $\mathbb{R}^3$ that is spanned by the following vectors

$$ (1,0,0) \ \ \ \ (1,1,0) \ \ \ \ (4,1,0) \ \ \ \ (0,-1,0)\ \ \ \ $$

I tried similar problems that reduced to similar dimensions (Screenshot attached below) but I can't seem to be able to solve this question. Could someone help me solve this problem and tell me why I was wrong in the first two attempts?

Screenshots here:

Attempt 2

Attempt 1][1]

My work is quite messy I'll try to add it. But I essentially created the augmented matrix, reduced it to RREF, and found that two vectors could be express by the two other vectors. For Q1, $v3_ = 3(v_1) + 1(v_2)$ and $v_4 = 2(v_1) – 2(v_2)$. For Q2, $v_3 = v_1 + v_2$ and $v_4 = v_1 – v_2$.

Best Answer

For Q1 and Q2, all the options suggest that the dimension of the subspace is $2$. As long as the two vectors are not multiple of each other, they can form a basis.

Remember that basis is not unique and we should not stop at the first correct answer.

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