One may get confused when asked to specify the coordinates of T as depicted below.

**Figure 1**

However, they may think it easy to determine the coordinates of T if instead of Figure 1, they are given the following figure.

**Figure 2**

As Figure 2 shows, the coordinates of T are (1,1). However, T does not have unique coordinates. In fact, the coordinates (in a two dimensional space) depend on where we put the point of reference O and a pair of nonparallel “yardsticks”. In Figure 3, the coordinates of T are (2,2) because we have used yardsticks whose lengths are half of the previous ones.

**Figure 3**

Loosely speaking, the coordinates of T are affected not only by the “unit” lengths of the yardsticks but also their directions.

**Figure 4**

In Figure 4, although the unit lengths of the yardsticks being used are equal, the coordinates of T still depend on whether we use the xy-plane or x’y’-plane. On the xy-plane, the coordinates of T is (2,2), while on the x’y’-plane its coordinates are . In linear algebra, we adopt the term * basis* (

*of a vector space*) to mean the same thing as yardsticks in the previous discussion. In brief, we claim that the coordinates of T above depend on the basis being used in the vector space.

Before looking at the definition of the coordinates of a vector in a vector space V, it is worth pointing out that every vector can be expressed as a unique linear combination of some fixed basis vectors, as described more precisely in the following theorem.

**The Unique Representation Theorem**

Let be a basis for a vector space V. Then for each in V, there exists a unique set of scalars *c*_{1}, *c*_{2}, …, *c*_{n} such that:

**Example 1**

Consider the vector and the basis where and . Find *c*_{1} and *c*_{2} such that .

**Answer**

Let c_{1} and c_{2} be real numbers satisfying . Consequently,

So we have a system of linear equations with two unknowns *c*_{1} and *c*_{2}. By applying elimination method (or other techniques to solve such problems), it can be shown that *c*_{1} = 3 and *c*_{2} = 1 satisfy .

**Example 2**

Let P_{n} be the vector space whose members, or vectors, are all real polynomial functions having degree ≤ n, that is, all functions expressible in the form p(x) = a_{0} + a_{1}x + … + a_{n}x^{n} where a_{0}, a_{1}, …, a_{n} are real numbers. A basis for P_{2} is where

Given , find *c*_{1}, *c*_{2}, and *c _{3}* such that .

**Answer**

6 + 8x – 2x^{2} = *c*_{1}(2 + x) + *c*_{2}(-1 + 3x + x^{2}) + *c*_{3}(-5x + 2x^{2})

6 + 8x – 2x^{2} = (2*c*_{1} – *c*_{2}) + (*c*_{1} + 3*c*_{2} – 5*c*_{3})x + (*c*_{2} + 2*c*_{3})x^{2}

From this, we have the following system of linear equations in *c*_{1}, *c*_{2}, and *c*_{3}.

By applying the Cramer’s Rule, for example, it can be shown that *c*_{1} = 3, *c*_{2} = 0, and *c*_{3} = -1 satisfy .

**Definition**

Let be a basis for a finite dimensional vector space V and be the expression for in terms of the basis B. Then the scalars *c*_{1}, *c*_{2}, …, *c*_{n} are called the * coordinates* of relative to the basis B. The

*of relative to B is denoted by and is the vector in defined by . The*

**coordinate vector***of relative to B is denoted by and is the n×1 matrix defined by*

**coordinate matrix**

**Example 3**

Find the coordinate vector and coordinate matrix of relative to the basis B in Example 1.

**Answer**

It has been shown in Example 1 that *c*_{1} = 3 and *c*_{2} = 1 satisfy . Thus,

By the notation just-defined above, the coordinate vector of relative to B is and its coordinate matrix (relative to B) is

**Example 4**

Find the coordinate vector and coordinate matrix of relative to the basis B in Example 2.

**Answer**

It has been shown in Example 2 that *c*_{1} = 3, *c*_{2} = 0, and *c*_{3} = -1 satisfy . Thus,

6 + 8x – 2x^{2} = 3(2 + x) + 0(-1 + 3x + x^{2}) + (-1)(-5x + 2x^{2})

By the notation just-defined above, the coordinate vector of relative to B is and its coordinate matrix (relative to B) is

**Example 5**

Referring to Example 2, suppose that we use the basis instead of B, where

Find the coordinate matrix of relative to the “new” basis C.

**Answer**

By applying the similar method as in Example 2 to find *c*_{1}, *c*_{2}, and *c*_{3}, it can be proved that *c*_{1} = 1, *c*_{2} = 1, and *c*_{3} = 1 satisfy . Thus,

6 + 8x – 2x^{2} = 1(1 + 2x – x^{2}) + 1(6x – 4x^{2}) + 1(5 + 3x^{2})

By the notation just-defined above, the coordinate matrix of relative to C is

It is of interest to know how relates to . Alternatively, how to express in terms of ? The next theorem answers the question.

**Theorem**

Suppose that and are bases for a vector space V and ∈ V. Then where the column vectors of P are .

Note:

The matrix P above can be denoted by and is called the * transition matrix *from C to B.

**Example 6**

Consider the vector and the following bases for :

and

- Find the coordinate vector and coordinate matrix of relative to B.
- By using the transition matrix P as described in the theorem above, determine the coordinate matrix of relative to C, i.e. .
- If
*c*_{1}and*c*_{2}are the first and the second coordinates of , respectively, verify that , where and .

**Answer**

Part a

By inspection, . Thus the coordinate vector of relative to B is . Consequently, its coordinate matrix is .

Part b

Let and . Since and , it follows that the coordinate matrices of and relative to B are and , respectively. Thus, the transition matrix from C to B is . The theorem above states that . Multiplying both sides of the equation by P^{-1} yields . Note that . So, the coordinate matrix of relative to C can be computed as follows.

Part c

Because* c*_{1} and *c*_{2} are the first and the second coordinates of , *c*_{1} = 7 and *c*_{2 }= -5. Consequently,

.

So, is verified.

In part b of Example 3, we have used the following corollary to the previous theorem:

Suppose that and are bases for a vector space V and ∈ V. Then where the column vectors of P are .

**Example 7**

Referring to Example 2, 4, and 5 above, consider the vector ∈ P_{2}. In Example 4, it is shown that . Using the transition matrix P as described in the theorem above, determine the coordinate matrix of relative to C, i.e. . Verify that we get the same result as obtained in Example 5.

**Answer**

The column vectors of P are . It can be shown that:

Therefore,

These produce the transition matrix P as follows.

It can be proved that .

To find , apply the corollary above. This yields:

The result is the same as obtained in Example 5.