matrix representation of a linear transformation

Linear transformations and matrices are the two most fundamental notions in the study of linear algebra. The two concepts are intimately related. In this article, we will see how the two are related. We assume that all vector spacesMathworldPlanetmath are finite dimensional and all vectors are written as column vectorsMathworldPlanetmath.

Linear transformations as matrices

Let V,W be vector spaces (over a common field k) of dimensionPlanetmathPlanetmath n and m respectively. Fix bases A={v1,,vn} and B={w1,,wm} for V and W respectively. We shall order these bases so that vi<vj and wi<wj whenever i<j. To distinguish an ordinary set from an ordered set, we shall adopt the notation v1,,vn to mean the set {v1,,vn} with ordering vivj whenever ij. The importance of ordering these bases will be apparent shortly.

For any linear transformation T:VW, we can write


for each j{1,,n} and αijk. We define the matrix associated with the linear transformation T and ordered bases A,B by


where 1in and 1jm. [T]BA is a m×n matrix whose entries are in k. When A=B, we often write [T]A:=[T]AA. In additionPlanetmathPlanetmath, when both ordered bases are standard bases En,Em ordered in the obvious way, we write [T]:=[T]EmEn.


  1. 1.

    Let T:34 be given by


    Using the standard ordered bases

    E3=(100),(010),(001) for 3 and E4=(1000),(0100),(0010),(0001) for 4

    ordered in the obvious way. Then,


    so the matrix [T]E4E3 associated with T and the standard ordered bases E3 and E4 is the 4×3 matrix

  2. 2.

    Let T be the same linear transformation as above. However, let E3 be the same basis as E3 except that the order is reversed: e3<e2<e1. Then


    Note that this matrix is just the matrix from the previous example except that the first and the last columns have been switched.

  3. 3.

    Again, let T be the same as before. Now, let E4 be the ordered basis whose elements are those of E4 but the order is now given by e2<e1<e4<e3. Then


    Note that this matrix is just the matrix from the previous example except that the first two rows and the last two rows have been interchanged.


  • From the examples above, we note several important features of a matrix representation of a linear transformation:

    1. (a)

      the matrix depends on the bases given to the vector spaces

    2. (b)

      the ordering of a basis is important

    3. (c)

      switching the order of a given basis amounts to switching columns and rows of the matrix, essentially multiplying a matrix by a permutation matrixMathworldPlanetmath.

  • Some basic properties of matrix representations of linear transformations are

    1. (a)

      If T:VW is a linear transformation, then [rT]BA=r[T]BA, where A,B are ordered bases for V,W respectively.

    2. (b)

      If S,T:VW are linear transformations, then [S+T]BA=[S]BA+[T]BA, where A and B are ordered bases for V and W respectively.

    3. (c)

      If S:UV and T:VW, then [TS]CA=[T]CB[S]BA, where A,B,C are ordered bases for U,V,W respectively.

    4. (d)

      As a result, T is invertiblePlanetmathPlanetmathPlanetmath iff [T]BA is an invertible matrix iff dim(V)=dim(W).

  • We could have represented all vectors as row vectors. However, doing so would mean that the matrix representation M1 of a linear transformation T would be the transposeMathworldPlanetmath of the matrix representation M2 of T if the vectors were represented as column vectors: M1=M2T, and that the application of the matrices to vectors would be from the right of the vectors:

    (abc)(10-13201011-52)   instead of   (121001-11-5302)(abc).

Matrices as linear transformations

Every m×n matrix A over a field k can be thought of as a linear transformation from kn to km if we view each vector vkn as a n×1 matrix (a column) and the mapping is done by the matrix multiplicationMathworldPlanetmath Av, which is a m×1 matrix (a column vector in km). Specifically, we define TA:knkm by


It is easy to see that TA is indeed a linear transformation. Furthermore, [TA]=[TA]EmEn=A, since the representation of vectors as n-tuples of elements in k is the same as expressing each vector under the standard basis (ordered) in the vector space kn. Below we list some of the basic properties:

  1. 1.

    TrA=rTA, for any rk,

  2. 2.

    TA+TB=TA+B, where A,B are m×n matrices over k

  3. 3.

    TATB=TAB, where A is an m×n matrix and B is an n×p matrix over k

  4. 4.

    TA is invertible iff A is an invertible matrix.

Remark. As we can see from the discussion above, if we fix sets of base elements for a vector space V and W, there is a one-to-one correspondence between the set of matrices (of the same size) over the underlying field k and the set of linear transformations from V to W.


  • 1 Friedberg, Insell, Spence. Linear Algebra. Prentice-Hall Inc., 1997.
Title matrix representation of a linear transformation
Canonical name MatrixRepresentationOfALinearTransformation
Date of creation 2013-03-22 17:29:59
Last modified on 2013-03-22 17:29:59
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 15
Author CWoo (3771)
Entry type Definition
Classification msc 15A04
Synonym ordered bases
Synonym standard ordered bases
Related topic LinearTransformation
Defines ordered basis
Defines matrix representation
Defines standard ordered basis