The standard basis for ${ℝ}^2$ is $\{{\mathbf{e}}_1,{\mathbf{e}}_2\}$. Any vector x in ${ℝ}^2$ can be expressed as a linear combination:

The scalars $x_1$ and $x_2$ can be thought of as the coordinates of x with respect to the standard basis. Actually, for any basis $\{\mathbf{y},\mathbf{z}\}$ for ${ℝ}^2$, it follows from Theorem 3.3.2 that a given vector x can be represented uniquely as a linear combination:

The scalars $\alpha$ and $\beta$ are the coordinates of x with respect to the basis $\{\mathbf{y},\mathbf{z}\}$. Let us order the basis elements so that y is considered the first basis vector and z is considered the second, and denote the ordered basis by $[\mathbf{y},\mathbf{z}]$. We can then refer to the vector ${(\alpha ,\beta )}^T$ as the coordinate vector of x with respect to $[\mathbf{y},\mathbf{z}]$. Note that, if we reverse the order of the basis vectors and take $[\mathbf{z},\mathbf{y}]$, then we must also reorder the coordinate vector. The coordinate vector of x with respect to $[\mathbf{z},\mathbf{y}]$ will be ${(\beta ,\alpha )}^T$. When we refer to a basis using subscripts, such as $\{{\mathbf{u}}_1,{\mathbf{u}}_2\}$, the subscripts assign an ordering to the basis vectors.

Figure 3.5.1.

As an example of a problem for which it is helpful to change coordinates, consider the following application.

Once we have decided to work with a new basis, we have the problem of finding the coordinates with respect to that basis. Suppose, for example, that instead of using the standard basis $\{{\mathbf{e}}_1,{\mathbf{e}}_2\}$ for ${ℝ}^2$, we wish to use a different basis, say,

Indeed, we may want to switch back and forth between the two coordinate systems. Let us consider the following two problems:

1. Given a vector $\text{x}={(x_1,x_2)}^T$, find its coordinates with respect to ${\mathbf{u}}_1$ and ${\mathbf{u}}_2$.

2. Given a vector $c_1{\mathbf{u}}_1+c_2{\mathbf{u}}_2$, find its coordinates with respect to ${\mathbf{e}}_1$ and ${\mathbf{e}}_2$.

We will solve II first, since it turns out to be the easier problem. To switch bases from $\{{\text{u}}_1,{\text{u}}_2\}$ to $\{{\mathbf{e}}_1,{\mathbf{e}}_2\}$, we must express the old basis elements ${\mathbf{u}}_1$ and ${\mathbf{u}}_2$ in terms of the new basis elements ${\mathbf{e}}_1$ and ${\mathbf{e}}_2$.

It follows then that

Thus, the coordinate vector of $c_1{\mathbf{u}}_1+c_2{\mathbf{u}}_2$ with respect to $\{{\mathbf{e}}_1,{\mathbf{e}}_2\}$ is

If we set

then, given any coordinate vector c with respect to $\{{\mathbf{u}}_1,{\mathbf{u}}_2\}$, to find the corresponding coordinate vector x with respect to $\{{\mathbf{e}}_1,{\mathbf{e}}_2\}$, we simply multiply U times c:

The matrix U is called the transition matrix from the ordered basis $\{{\mathbf{u}}_1,{\mathbf{u}}_2\}$ to the standard basis $\{{\mathbf{e}}_1,{\mathbf{e}}_2\}$.

To solve problem I, we must find the transition matrix from $\{{\mathbf{e}}_1,{\mathbf{e}}_2\}$ to $\{{\mathbf{u}}_1,{\mathbf{u}}_2\}$. The matrix U in (2) is nonsingular, since its column vectors, ${\mathbf{u}}_1$ and ${\mathbf{u}}_2$, are linearly independent. It follows from (2) that

Thus, given a vector

we need only multiply by $U^{-1}$ to find its coordinate vector with respect to $\{{\mathbf{u}}_1,{\mathbf{u}}_2\}$. $U^{-1}$ is the transition matrix from $\{{\mathbf{e}}_1,{\mathbf{e}}_2\}$ to $\{{\mathbf{u}}_1,{\mathbf{u}}_2\}$.

Now let us consider the general problem of changing from one ordered basis $\{{\mathbf{v}}_1,{\mathbf{v}}_2\}$ of ${ℝ}^2$ to another ordered basis $\{{\mathbf{u}}_1,{\mathbf{u}}_2\}$. In this case, we assume that, for a given vector x, its coordinates with respect to $\{{\mathbf{v}}_1,{\mathbf{v}}_2\}$ are known:

Now we wish to represent x as a sum $d_1{\text{u}}_1+d_2{\text{u}}_2$. Thus, we must find scalars $d_1$ and $d_2$ so that

If we set $V=({\text{v}}_1,{\text{v}}_2)$ and $U=({\mathbf{u}}_1,{\mathbf{u}}_2)$, then Equation (3) can be written in matrix form:

It follows that

Thus, given a vector x in ${ℝ}^2$ and its coordinate vector c with respect to the ordered basis $\{{\mathbf{v}}_1,{\mathbf{v}}_2\}$, to find the coordinate vector of x with respect to the new basis $\{{\mathbf{u}}_1,{\mathbf{u}}_2\}$, we simply multiply c by the transition matrix $S=U^{-1}V$.

The change of basis from $\{{\mathbf{v}}_1,{\mathbf{v}}_2\}$ to $\{{\mathbf{u}}_1,{\mathbf{u}}_2\}$ can also be viewed as a two-step process. First we change from $\{{\mathbf{v}}_1,{\mathbf{v}}_2\}$ to the standard basis, $\{{\mathbf{e}}_1,{\mathbf{e}}_2\}$, and then we change from the standard basis to $\{{\mathbf{u}}_1,{\mathbf{u}}_2\}$. Given a vector x in ${ℝ}^2$, if c is the coordinate vector of x with respect to $\{{\mathbf{v}}_1,{\mathbf{v}}_2\}$ and d is the coordinate vector of x with respect to $\{{\mathbf{u}}_1,{\mathbf{u}}_2\}$, then

Since V is the transition matrix from $\{{\mathbf{v}}_1,{\mathbf{v}}_2\}$ to $\{{\mathbf{e}}_1,{\mathbf{e}}_2\}$ and $U^{-1}$ is the transition matrix from $\{{\mathbf{e}}_1,{\mathbf{e}}_2\}$ to $\{{\mathbf{u}}_1,{\mathbf{u}}_2\}$, it follows that

and hence

As before, we see that the transition matrix from $\{{\mathbf{v}}_1,{\mathbf{v}}_2\}$ to $\{{\mathbf{u}}_1,{\mathbf{u}}_2\}$ is $U^{-1}V$ (see Figure 3.5.2).

Figure 3.5.2.

Everything we have done so far can easily be generalized to apply to any finite-dimensional vector space. We begin by defining coordinate vectors for an n-dimensional vector space.

The examples considered so far have all dealt with changing coordinates in ${ℝ}^2$. Similar techniques could be used for ${ℝ}^n$. In the case of ${ℝ}^n$, the transition matrices will be $n\times n$.

If V is any n-dimensional vector space, it is possible to change from one basis to another by means of an $n\times n$ transition matrix. We will show that such a transition matrix is necessarily nonsingular. To see how this is done, let $E=\{{\mathbf{w}}_1,\dots ,{\mathbf{w}}_n\}$ and $F=\{{\mathbf{v}}_1,\dots ,{\mathbf{v}}_n\}$ be two ordered bases for V. The key step is to express each basis vector ${\mathbf{w}}_j$ as a linear combination of the ${\mathbf{v}}_i\text{’s}$.

Let $\mathbf{v}\in V$. If $\mathbf{x}=[\mathbf{v}{]}_E$, it follows from (4) that

Thus, if $\mathbf{y}=[\mathbf{v}{]}_F$, then

and hence,

The matrix S defined by (4) is referred to as the transition matrix. Once S has been determined, it is a simple matter to change coordinate systems. To find the coordinates of $\text{v}=x_1{\text{w}}_1+\cdots +x_n{\text{w}}_n$ with respect to $\{{\mathbf{v}}_1,\dots ,{\mathbf{v}}_n\}$, we need only calculate $\mathbf{y}=S\mathbf{x}$.

The transition matrix S corresponding to the change of basis from $\{{\mathbf{w}}_1,\dots ,{\mathbf{w}}_n\}$ to $\{{\mathbf{v}}_1,\dots ,{\mathbf{v}}_n\}$ can be characterized by the condition

Taking $\mathbf{y}=0$ in (5), we see that $S\mathbf{x}=0$ implies that

Since the ${\mathbf{w}}_i\text{’s}$ are linearly independent, it follows that $\mathbf{x}=\mathbf{0}$. Thus, the equation $S\mathbf{x}=\mathbf{0}$ has only the trivial solution and hence the matrix S is nonsingular. The inverse matrix is characterized by the condition

Thus, $S^{-1}$ is the transition matrix used to change basis from $\{{\mathbf{v}}_1,\dots ,{\mathbf{v}}_n\}$ to $\{{\mathbf{w}}_1,\dots ,{\mathbf{w}}_n\}$.

We have seen that each transition matrix is nonsingular. Actually, any nonsingular matrix can be thought of as a transition matrix. If S is an $n\times n$ nonsingular matrix and $\{{\mathbf{v}}_1,\dots ,{\mathbf{v}}_n\}$ is an ordered basis for V, then define $\{{\mathbf{w}}_1,{\mathbf{w}}_2,\dots ,{\mathbf{w}}_n\}$ by (4). To see that the ${\text{w}}_j\text{’s}$ are linearly independent, suppose that

It follows from (4) that

By the linear independence of the ${\mathbf{v}}_i\text{’s}$, it follows that

or, equivalently,

Since S is nonsingular, x must equal 0. Therefore, ${\mathbf{w}}_1,\dots ,{\mathbf{w}}_n$ are linearly independent and hence they form a basis for V. The matrix S is the transition matrix corresponding to the change from the ordered basis $\{{\mathbf{w}}_1,\dots ,{\mathbf{w}}_n\}$ to $\{{\mathbf{v}}_1,\dots ,{\mathbf{v}}_n\}$.

In many applied problems, it is important to use the right type of basis for the particular application. In Chapter 5, we will see that the key to solving least squares problems is to switch to a special type of basis called an orthonormal basis. In Chapter 6, we will consider a number of applications involving the eigenvalues and eigenvectors associated with an $n\times n$ matrix A. The key to solving these types of problems is to switch to a basis for ${ℝ}^n$ consisting of eigenvectors of A.