Ordered bases

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

Let $\{v_1,\dots ,v_n\}$ be a basis for $V$. We call $(v_1,\dots ,v_n)$ 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 ${\lambda }_1,\dots ,{\lambda }_n\in \mathbb{F}$ such that $v={\lambda }_1{v}_1+\cdots +{\lambda }_n{v}_n.$

Define $[v{]}_{(v_1,\dots ,v_n)}$ to be the $n$-tuple $\left[\begin{array}{c}{\lambda }_1\\ {\lambda }_2\\ ⋮\\ {\lambda }_n\end{array}\right]$.

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

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

   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 $u=\alpha \left[\begin{array}{c}1\\ -1\end{array}\right]+\beta \left[\begin{array}{c}2\\ 3\end{array}\right]$. Thus, we need to solve the system $$\begin{array}{r}3=\alpha +2\beta \\ 2=-\alpha +3\beta \end{array}$$ Solving gives $\alpha =1$ and $\beta =1$. Hence, $[u{]}_{\mathrm{\Gamma }}=\left[\begin{array}{c}1\\ 1\end{array}\right]$.

2. Let $V$ denote the vector space of polynomials in $x$ with real coefficients having degree at most $2$. Let $\mathrm{\Gamma }$ denote the ordered basis $(1,x+1,x^2+1)$ and let $\mathrm{\Omega }$ 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 $[x{]}_{\mathrm{\Gamma }}$, 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 $[x{]}_{\mathrm{\Gamma }}=\left[\begin{array}{c}-1\\ 1\\ 0\end{array}\right]$.

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

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 $\mathrm{\Gamma }$ is an ordered basis for $V$, then one can easily check that $[u+v{]}_{\mathrm{\Gamma }}=[u{]}_{\mathrm{\Gamma }}+[v{]}_{\mathrm{\Gamma }}.$ 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 $[\lambda u{]}_{\mathrm{\Gamma }}=\lambda [u{]}_{\mathrm{\Gamma }}$. Thus, multiplying a vector by a scalar simply involves multiplying the corresponding tuple representation by the same scalar.

Quick Quiz

Exercises

1. Let $\mathrm{\Gamma }=(\left[\begin{array}{c}1\\ 2\end{array}\right],\left[\begin{array}{c}2\\ 3\end{array}\right])$ and $\mathrm{\Omega }=(\left[\begin{array}{c}1\\ -1\end{array}\right],\left[\begin{array}{c}1\\ -2\end{array}\right])$ be two ordered basis for ${\mathbb{R}}^2$.

   1. Let $u=\left[\begin{array}{c}1\\ 0\end{array}\right]$ and $v=\left[\begin{array}{c}0\\ 1\end{array}\right]$. Find the following:

      1. $[u{]}_{\mathrm{\Gamma }}$  

         $\left[\begin{array}{c}-3\\ 2\end{array}\right]$

      2. $[u{]}_{\mathrm{\Omega }}$  

         $\left[\begin{array}{c}2\\ -1\end{array}\right]$

      3. $[v{]}_{\mathrm{\Gamma }}$  

         $\left[\begin{array}{c}2\\ -1\end{array}\right]$

      4. $[v{]}_{\mathrm{\Omega }}$  

         $\left[\begin{array}{c}1\\ -1\end{array}\right]$

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

      $\left[\begin{array}{cc}4& 7\\ -3& -5\end{array}\right]$