Need help understanding the Basis Theorem and applying it in computation
The Basis theorem is stated as follows: Let $V$ be a $p$-dimensional vector space where $p$ is equal to or greater than $1$. Any linearly independent set of exactly $P$ elements in $V$ is automatically a basis for $V$. Any set of exactly $P$ elements that spans $V$ is automatically a basis for $V$.
My question is what is $P$? Could $p$ be referring to the rank of a matrix? If I a given a set of two vectors and they are linearly independent would that make the set a basis, since it has two vectors and to be in $\mathbb R^2$ it must have at least $2$ vectors. So it could be in $\mathbb R^2$ but by showing it is consistent it would prove that it spans $\mathbb R^2$.
Now applying it to problems on my final:
Show that $$S=\{u_1,u_2\}$$ where $$ u_1= \begin{pmatrix} 1 \\ 1\\ \end{pmatrix} $$ and $$ u_2= \begin{pmatrix} 1 \\ -1\\ \end{pmatrix} $$ is an orthoganl basis for $\mathbb R^2$.
I know that I can show it is orthogonal by showing the two vectors dot to 0. But How do I show that it is a basis using the above theorem?
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