## Ordered bases

Let $$V$$ be a vector space over the field $$\mathbb{F}$$ having dimension $$n$$.

Let $$\{v_1,\ldots,v_n\}$$ be a basis for $$V$$. We call $$(v_1,\ldots,v_n)$$ an ordered basis for $$V$$.

By fixing an order on the basis elements, we can now represent every vector in $$V$$ as an $$n$$-tuple.

More precisely, let $$v \in V$$. Recall that there exist scalars $$\lambda_1,\ldots,\lambda_n\in \mathbb{F}$$ such that $$v = \lambda_1 v_1 + \cdots + \lambda_n v_n.$$

Define $$[v]_{(v_1,\ldots,v_n)}$$ to be the $$n$$-tuple $$\begin{bmatrix} \lambda_1 \\ \lambda_2 \\ \vdots \\ \lambda_n\end{bmatrix}$$.

We call $$[v]_{(v_1,\ldots,v_n)}$$ a tuple representation (or coordinate representation of $$v$$ with respect to the ordered basis $$(v_1,\ldots,v_n)$$.

Clearly, if we choose a different basis or a different ordering of the basis vectors, we get a different representation for the same vector. That is why it is so important to specify the ordered basis when working with tuple representations.

### Examples

1. Note that $$\Gamma = \left ( \begin{bmatrix} 1 \\ -1 \end{bmatrix}, \begin{bmatrix} 2 \\ 3\end{bmatrix}\right )$$ is an ordered basis for $$\mathbb{R}^2$$. Let $$u = \begin{bmatrix} 3 \\ 2 \end{bmatrix}$$. What is $$[u]_\Gamma$$?

Solution. We first write $$u$$ as a linear combination of the ordered basis elements. That is, we want to find real numbers $$\alpha$$ and $$\beta$$ such that $$u = \alpha \begin{bmatrix} 1 \\ -1 \end{bmatrix} + \beta \begin{bmatrix} 2 \\ 3\end{bmatrix}$$. Thus, we need to solve the system \begin{eqnarray*}3 = \alpha + 2\beta \\ 2 = -\alpha + 3 \beta \end{eqnarray*} Solving gives $$\alpha = 1$$ and $$\beta = 1$$. Hence, $$[u]_\Gamma = \begin{bmatrix} 1 \\ 1\end{bmatrix}$$.

2. Let $$V$$ denote the vector space of polynomials in $$x$$ with real coefficients having degree at most $$2$$. Let $$\Gamma$$ denote the ordered basis $$(1, x+1, x^2+1)$$ and let $$\Omega$$ denote the ordered basis $$(x-1, x^2+1, x^2-1)$$. (Check that these are indeed ordered bases.) We want to find the tuple representations of $$x$$ with respect to these ordered bases.

To determine $$[x]_\Gamma$$, we need to find real numbers $$\alpha, \beta,\gamma$$ such that $x = \alpha (1) + \beta (x+1) + \gamma (x^2+1).$ Simplifying the right-hand side, we obtain $x = \gamma x^2 + \beta x + (\alpha + \beta + \gamma).$ Since the left-hand side has no $$x^2$$ term, we must have $$\gamma = 0$$. Now, comparing the coefficients of $$x$$ on both sides, we get $$\beta = 1$$. Since the left-hand side has no constant term, we must have $$\alpha = -1$$.

In other words, $$x = (-1)1 + 1(x+1) + 0x^2$$, giving $$[x]_\Gamma = \begin{bmatrix} -1 \\ 1 \\ 0\end{bmatrix}$$.

For $$[x]_\Omega$$, note that $$x = 1(x-1) + \frac{1}{2}(x^2+1) + \left(-\frac{1}{2}\right)(x^2)$$. Hence, $$[x]_\Omega = \begin{bmatrix} 1 \\ \frac{1}{2} \\ -\frac{1}{2}\end{bmatrix}$$.

## Working with tuple representations

The beauty of these tuple representations of vectors is that we can now work with tuples instead. For instance, if $$u,v\in V$$ and $$\Gamma$$ is an ordered basis for $$V$$, then one can easily check that $$[u+v]_{\Gamma} = [u]_{\Gamma} + [v]_{\Gamma}.$$ In other words, adding two vectors in $$V$$ simply requires adding the corresponding tuple representations in $$\mathbb{F}^n$$.

And if $$\lambda$$ is a scalar, then $$[\lambda u]_{\Gamma} = \lambda [u]_{\Gamma}$$. Thus, multiplying a vector by a scalar simply involves multiplying the corresponding tuple representation by the same scalar.

## Exercises

1. Let $$\Gamma = \left ( \begin{bmatrix} 1 \\ 2\end{bmatrix}, \begin{bmatrix} 2 \\ 3 \end{bmatrix}\right )$$ and $$\Omega = \left ( \begin{bmatrix} 1 \\ -1\end{bmatrix}, \begin{bmatrix} 1 \\ -2 \end{bmatrix}\right)$$ be two ordered basis for $$\mathbb{R}^2$$.
1. Let $$u = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$$ and $$v = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$$. Find the following:

1. $$[u]_{\Gamma}$$

2. $$[u]_{\Omega}$$

3. $$[v]_{\Gamma}$$

4. $$[v]_{\Omega}$$

2. Give a $$2 \times 2$$ matrix $$A$$ such that $$[x]_\Omega = A [x]_\Gamma$$ for all $$x \in \mathbb{R}^2$$.