Bases and Coordinates in R2
This applet demonstrates the concept of coordinate vectors in $\mathbb R^2$
A basis $\mathcal B$ of a vector space $V$ is a linearly independent spanning set.
One useful feature of a basis is that it gives rise to a way of writing coordinates on $V$. Any vector $\mathbf v \in V$ can be written uniquely as a linear combination of the basis vectors in $\mathcal B$. If $V$ has dimension $n$, any vector $\mathbf v$ can therefore be uniquely specified by a list of $n$ scalars: $$[\mathbf v]_{\mathcal B} = \begin{bmatrix}\alpha_1\\\vdots\\\alpha_n\end{bmatrix} \Leftrightarrow \mathbf v = \alpha_1 \mathbf b_1 + … + \alpha_n \mathbf b_n.$$ The applet shows a vector $\mathbf v$ drawn in black, as well as a blue coordinate grid which corresponds to a basis $\mathcal B$. You can click and drag the tips of the vectors to change the vector $\mathbf v$ or the basis vectors in $\mathcal B$. You can also enable a second coordinate grid corresponding to a different basis $\mathcal C$ by clicking the checkbox on the right.
As you adjust the vectors, the coordinate vector visible on the right of the applet will change. Is it consistent with what you visible in the applet? Count out the number of steps you need to take along the blue coordinate grid axes to reach the vector $\mathbf v$.
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