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Section3.1Matrix Transformations

  1. Learn to view a matrix geometrically as a function.
  2. Learn examples of matrix transformations: reflection, dilation, rotation, shear, projection.
  3. Understand the vocabulary surrounding transformations: domain, codomain, range.
  4. Understand the domain, codomain, and range of a matrix transformation.
  5. Pictures: common matrix transformations.
  6. Vocabulary words: transformation / function, domain, codomain, range, identity transformation, matrix transformation.

In this section we learn to understand matrices geometrically as functions, or transformations. We briefly discuss transformations in general, then specialize to matrix transformations, which are transformations that come from matrices.

Subsection3.1.1Matrices as Functions

Informally, a function is a rule that accepts inputs and produces outputs. For instance, f ( x )= x 2 is a function that accepts one number x as its input, and outputs the square of that number: f ( 2 )= 4. In this subsection, we interpret matrices as functions.

Let A be a matrix with m rows and n columns. Consider the matrix equation b = Ax (we write it this way instead of Ax = b to remind the reader of the notation y = f ( x ) ). If we vary x , then b will also vary; in this way, we think of A as a function with independent variable x and dependent variable b .

  • The independent variable (the input) is x , which is a vector in R n .
  • The dependent variable (the output) is b , which is a vector in R m .

The set of all possible output vectors are the vectors b such that Ax = b has some solution; this is the same as the column space of A by this note in Section 2.3.

R n R m b = Ax x Ax Col ( A )


At this point it is convenient to fix our ideas and terminology regarding functions, which we will call transformations in this book. This allows us to systematize our discussion of matrices as functions.


A transformation from R n to R m is a rule T that assigns to each vector x in R n a vector T ( x ) in R m .

  • R n is called the domain of T .
  • R m is called the codomain of T .
  • For x in R n , the vector T ( x ) in R m is the image of x under T .
  • The set of all images { T ( x ) | x in R n } is the range of T .

The notation T : R n −→ R m means T is a transformation from R n to R m .

It may help to think of T as a “machine” that takes x as an input, and gives you T ( x ) as the output.

R n R m domain codomain T x T ( x ) range T

The points of the domain R n are the inputs of T : this simply means that it makes sense to evaluate T on vectors with n entries, i.e., lists of n numbers. Likewise, the points of the codomain R m are the outputs of T : this means that the result of evaluating T is always a vector with m entries.

The range of T is the set of all vectors in the codomain that actually arise as outputs of the function T , for some input. In other words, the range is all vectors b in the codomain such that T ( x )= b has a solution x in the domain.


The identity transformation Id R n : R n R n is the transformation defined by the rule

Id R n ( x )= x forall x in R n .

In other words, the identity transformation does not move its input vector: the output is the same as the input. Its domain and codomain are both R n , and its range is R n as well, since every vector in R n is the output of itself.

Subsection3.1.3Matrix Transformations

Now we specialize the general notions and vocabulary from the previous subsection to the functions defined by matrices that we considered in the first subsection.


Let A be an m × n matrix. The matrix transformation associated to A is the transformation

T : R n −→ R m deBnedby T ( x )= Ax .

This is the transformation that takes a vector x in R n to the vector Ax in R m .

If A has n columns, then it only makes sense to multiply A by vectors with n entries. This is why the domain of T ( x )= Ax is R n . If A has n rows, then Ax has m entries for any vector x in R n ; this is why the codomain of T ( x )= Ax is R m .

The definition of a matrix transformation T tells us how to evaluate T on any given vector: we multiply the input vector by a matrix. For instance, let

A = I 123456 J

and let T ( x )= Ax be the associated matrix transformation. Then

T A 1 2 3 B = A A 1 2 3 B = I 123456 JA 1 2 3 B = I 14 32 J .

Suppose that A has columns v 1 , v 2 ,..., v n . If we multiply A by a general vector x , we get

Ax = A ||| v 1 v 2 ··· v n ||| BCGGE x 1 x 2 ... x n DHHF = x 1 v 1 + x 2 v 2 + ··· + x n v n .

This is just a general linear combination of v 1 , v 2 ,..., v n . Therefore, the outputs of T ( x )= Ax are exactly the linear combinations of the columns of A : the range of T is the column space of A . See this note in Section 2.3.

Let A be an m × n matrix, and let T ( x )= Ax be the associated matrix transformation.

  • The domain of T is R n , where n is the number of columns of A .
  • The codomain of T is R m , where m is the number of rows of A .
  • The range of T is the column space of A .

In the case of an n × n square matrix, the domain and codomain of T ( x )= Ax are both R n . In this situation, one can regard T as operating on R n : it moves the vectors around in the same space.