2.3 Dimension

what is a dimension

• if $b_1$ and $b_2$ are two different bases for a 1.4 Subspace S, then $b_1$and $b_2$ have the same number of vectors

• EX1 dimension of ${\mathbb{R}}^3$?

  • a basis for ${\mathbb{R}}^3$ :

$$(\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right])⟹dim({\mathbb{R}}^3)=3$$

• EX2 Dimension of the span $(1,1),(1,-1)$?

  • A basis:

$$(\left[\begin{array}{c}1\\ 1\end{array}\right],\left[\begin{array}{c}1\\ -1\end{array}\right])⟹dim=2$$