Basis vectors in 2 dimensions

A foundational concept in linear algebra is that of a basis. One can think of a basis as a finite set of vectors that can be used to describe any other vectors of the same dimension. For example:

$$\hat{i} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \quad \hat{j} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}.$$

form a basis for $\mathbb{R}^2$. 

Why?

Consider an arbitrary vector $\vec{v} \in \mathbb{R}^2$, which we can write as $\vec{v} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix}$.

Then:

$$\vec{v} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = v_1 \begin{bmatrix} 1 \\ 0 \end{bmatrix} + v_2 \begin{bmatrix} 0 \\ 1 \end{bmatrix} = v_1 \hat{i} + v_2 \hat{j}.$$

So we’ve expressed an arbitrary vector in terms of a linear combination of the set of vectors:

$$B = \{\hat{i}, \hat{j}\} = \left\{\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}\right\}.$$

So $B$ is a basis for the vector space $\mathbb{R}^2$. In fact, we call it the standard basis for $\mathbb{R}^2$. 

And actually, this works for $\mathbb{C}^2$ as well! Since the same argument holds if $v_1$ and $v_2$ are complex numbers or real numbers.

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