[Math] What exactly is Standard Coordinates

Question

What exactly is a standard coordinates?

Sorry it seems like a very stupid question, but my professor didn't really explain it and just started to use it for solving other problems related to transformation, so I got a little bit lost.

Does it mean the standard basis in $\mathbb{R}^n$? For example, vector $(1,0)$ and $(0,1)$ are the vectors form a basis for $\mathbb{R}^2$, does it mean $(1,0)\ ,\ (0,1)$ are the standard coordinates?

Could you please provide me an example of a standard coordinates?

Thank you very much!

Answer 1

"Standard coordinates" typically denotes coordinates with respect to the standard basis.

Consider the following definition.

If $\mathcal{A}=(a_1,...,a_n)$ and $\mathcal{B} = (b_1,...,b_n)$ are two bases of an $n$-dimensional linear space $V$, then the change of basis matrix from $\mathcal{B}$ to $\mathcal{A}$ is a matrix $S_{\mathcal{B}\rightarrow \mathcal{A}}$ such that:

$$[f]_{\mathcal{A}} = S[f]_{\mathcal{B}}, \forall f \in V$$

Note that:

$$S_{\mathcal{B}\rightarrow \mathcal{A}} = \left( \begin{array}{ccc} | & & |\\ [b_1]_\mathcal{A} & ... & [b_n]_\mathcal{A}\\ | & & |\end{array} \right)$$

So coming back to your question (in the comments): if $\mathcal{B} = \{\left( \begin{array}{ccc} 6 \\ -5 \end{array} \right),\left( \begin{array}{ccc} -7 \\ 4 \end{array} \right)\}$ is a basis of $\mathbb{R}^2$, then a matrix that converts from $\mathcal{B}$ coordinates to standard coordinates $\mathcal{E} = \{\left( \begin{array}{ccc} 1 \\ 0 \end{array} \right),\left( \begin{array}{ccc} 0 \\ 1 \end{array} \right)\}$ in $\mathbb{R}^2$ is:

$$S_{\mathcal{B}\rightarrow \mathcal{E}} = \left( \begin{array}{ccc} -\frac{4}{11}& -\frac{7}{11}\\ -\frac{5}{11}& -\frac{6}{11}\end{array} \right)$$

I'll leave it to you to verify this result.