The central proposition in this article is that for every linear transformation from a vector space to another there exists a matrix that can be viewed as a representation of the transformation and the entries of the matrix depend on the bases chosen for the vector spaces. As a motivating example, consider the linear transformation T from to defined by T((x_{1},x_{2},x_{3})) = (2x_{1}-x_{2}+x_{3},x_{1}+3x_{2}-2x_{3}). The transformation can be also expressed as

Here is the coordinate matrix of the vector (x_{1},x_{2},x_{3}) relative to some predetermined basis B_{1}, i.e. . The expression denotes the coordinate matrix of the vector T((x_{1},x_{2},x_{3})) relative to some predetermined basis B_{2}. In the motivating example, the bases chosen for each vector space are standard bases, that is B_{1} = {(1,0,0), (0,1,0), (0,0,1)} for and B_{2} = {(1,0), (0,1)} for . In the example, is called the matrix of the linear transformation T with respect to the bases B_{1} and B_{2}.

The questions reasonably arising are as follows.

- Is there always such matrix as A above for every linear transformation?
- If there is, how to find A?

The theorem below addresses the questions.

**Theorem**

Let V, W be vector spaces. Let *B* and *C* be the bases for V and W respectively. If T is a linear transformation from V to W then there exists a matrix A such that for every ∈ V. Moreover, if V is of dimension n with basis and W is of dimension m with basis then A is an m×n matrix which is determined as follows.

If ∈ V and then .

The matrix A as determined above is called the matrix of T with respect to the bases B and C.

**Example 1**

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. Let T be the linear transformation from P_{1} to P_{2} defined as follows.

T((a_{0} + a_{1}x)) = (a_{0} – a_{1}) + (2a_{0} – 3a_{1})x + (a_{0} + a_{1})x^{2}

Given the standard bases B = {1, x} and C = {1, x, x^{2}} for P_{1} and P_{2}, respectively, find the matrix of T with respect to the bases B and C.

**Answer**

Let where . By the definition of T, it follows that

Since the basis for P_{2} is the standard basis C = {1, x, x^{2}}, the coordinate matrices of and can be easily obtained as follows.

and

The theorem above implies that the matrix of T with respect to the bases B and C is .

**Example 2**

Referring to Example 1, suppose that .

- Find the coordinate matrix of relative to the standard basis C by computing .
- Using obtained in part a, compute .
- Find by the definition of T, that is T((a
_{0}+ a_{1}x)) = (a_{0}– a_{1}) + (2a_{0}– 3a_{1})x + (a_{0}+ a_{1})x^{2}and verify that the result is the same as the one obtained in part b.

**Answer**

Part a

Note that . Consequently, the coordinate matrix of relative to B is and the coordinate matrix of relative to the standard basis C is

.

Part b

By the definition of the coordinate matrix of a vector relative to a basis, implies .

Part c

By the definition of T, the image of (a_{0} + a_{1}x) under T is (a_{0} – a_{1}) + (2a_{0} – 3a_{1})x + (a_{0} + a_{1})x^{2}.

Thus, = T((2 + 5x)) = (2-5) + (2⋅2 – 3⋅5)x + (2 + 5)x^{2} = -3 – 11x + 7x^{2}. This agrees with the result in part b.

That the matrix of a linear transformation depends on the basis chosen for each vector space is illustrated in Example 3 below.

**Example 3**

Let T be the linear transformation from to (i.e., T is a linear operator on ) defined by T((x,y)) = (3x+5y,3x+y). Some possible bases for are the standard basis B = {(1,0), (0,1)} and C = {(-1,1), (5,3)}.

- Find the matrix of T with respect to the B. (Name the matrix K.)
- Find the matrix of T with respect to the C. (Name the matrix L.)
- Let . Find from and and compare the results.

**Answer**

Part a

Let . Then

Since the basis chosen is the standard basis, it is easy to deduce that and .

By the theorem above, the required matrix is .

Part b

Let . Then

Note that . From this, we have .

Similarly, , from which we get .

By the theorem above, the desired matrix is .

Part c

Finding from :

Thus, .

Finding from :

It can be easily verified that . Accordingly, .

This gives . The result is the same as obtained previously.

**Example 4**

Let T be the linear operator on P_{1} defined as follows.

T((a_{0} + a_{1}x)) = (2a_{0} – a_{1}) + (4a_{0} + 3a_{1})x

Some possible bases for P_{1} are the standard basis B = {1, x} and C = {(2+x), (-1+3x)}.

- Find the matrix of T with respect to the B. (Name the matrix K.)
- Find the matrix of T with respect to the C. (Name the matrix L.)
- Let . Find from and and compare the results.

**Answer**

Let . Then

Since the basis chosen is the standard basis, it is easy to deduce that and .

By the theorem above, the required matrix is .

Part b

Let . Then

It can be verified that . This results in .

Similarly, , from which we get .

By the theorem above, the desired matrix is .

Part c

Since , it follows that and .

Using K, we have:

Using L, we have:

The two formulae/methods yield the same result.

**Example 5**

Let T be the linear operator on P_{1} defined by T(a_{0} + a_{1}x) = (3a_{0}+2a_{1}) + (a_{0} – 4a_{1})x. Two bases for P_{1} are and where and .

- Find the matrix A such that for every ∈ P
_{1}. (In this article, such matrix is referred to aswith respect to some fixed bases. As for how to interpret this matrix, see Example 6.)**inter-bases matrix of a linear operator** - Given ∈ P
_{1}, find using . - Compare the result in part b to the one obtained by direct computation by substituting a
_{0}= 15 and a_{1}= -35 into T(a_{0}+ a_{1}x) = (3a_{0}+2a_{1}) + (a_{0}– 4a_{1})x.

**Answer**

Part a

Thus,

.

This gives .

By the theorem above, . So, the required matrix is .

Part b

Note that . So, . Consequently,

Part c

= T(15-35x) = (3⋅15 + 2⋅(-35)) + (15 – 4⋅(-35))x = -25 + 155x.

The result obtained in this part is consistent with the one obtained in part b.

**Example 6 (Interpretation of Inter-bases Matrix of A Linear Operator)**

Let T be the linear operator on P_{1} defined by T(a_{0} + a_{1}x) = (3a_{0}+2a_{1}) + (a_{0} – 4a_{1})x. Two bases for P_{1} are and where and . It can be shown that for every ∈ P_{1}, where .

- Given ∈ P
_{1}with satisfying , find . - Using the inter-bases matrix A in Example 5, find and compare the results.

**Answer**

Part a

We first have to find the transition matrix from B to C. It can be shown that and . Consequently, the transition matrix from B to C is . It follows that

Part b

In Example 5, . As shown in part b of the example, . The two methods result in identical results. The inter-bases matrix serves as a ‘shortcut’ when the matrix of a linear operator is formulated based on a basis which differs from the basis used in constructing the coordinate matrix of an input vector.