Describe the span of the set consisting of these two points in $\mathbb{R}^2$

Question

I just started my linear algebra course and I came across this question.

Describe the span of the set consisting of these two points in $\mathbb R^2:e_1=(1,0),e_2=(0,1)$

Definition of span: Span of a set of vectors is the collection of all linear combinations of vectors in the given set.

Solution in my textbook is given as: $\alpha e_1+\beta e_2=a$$ \begin{bmatrix} 1 \\ 0 \\ \end{bmatrix} +\beta \begin{bmatrix} 0 \\ 1 \\ \end{bmatrix}= \begin{bmatrix} \alpha \\ \beta \\ \end{bmatrix}$

My doubt is why did we write vector $e_1$ as $\begin{bmatrix} 1 \\ 0 \\ \end{bmatrix}$(Column matrix) but not $ \begin{bmatrix}1&2\\ \end{bmatrix} $(Row matrix)

Also please give me some more knowledge or tips before starting these concepts.

Answer 1

Since the span is $\alpha\begin{bmatrix} 1 \\ 0 \\ \end{bmatrix} +\beta \begin{bmatrix} 0 \\ 1 \\ \end{bmatrix}= \begin{bmatrix} \alpha \\ \beta \\ \end{bmatrix}$ where $\alpha,\beta\in\mathbb{R}$ can be chosen arbitrarily, the span will be $\mathbb{R}^2$ itself.

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Column matrix notation is the one you want to use when also considering matrices, since you usually write matrix vector multiplication like $\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}\begin{bmatrix} e \\ f \\ \end{bmatrix}$. If you don't consider matrices, it doesn't really matter if you represent $e_1$ as column matrix $\begin{bmatrix} 1 \\ 0 \\ \end{bmatrix}=(1,0)^T$ or row matrix (actually more like coordinate) $(1,0)$.