I need a little help with this question: Find a basis and dimension of the subspace R^n spanned by the following set:
{(1,3),(-1,2),(7,6)} (n=2)
I have tried attempting the question by putting the vectors in an augmented matrix and reducing it to row echelon form followed by taking the leading entries as the basis. I got {(1,3),(-1,2)} as my answer for the basis and 2 as the dimension. However, the solution provided {(1,0),(0,1)} as the answer for the basis.
I do not know where I'm going wrong with my answer. I would really appreciate it if I could get a little help on this question.
Thank You! :)
Both answers are correct. The space spanned by those vectors is $\mathbb{R}^2$, and both $\bigl((1,0),(0,1)\bigr)$ and $\bigl((1,3),(-1,2)\bigr)$ are bases of $\mathbb{R}^2$.