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 spaces are finite dimensional and all vectors are written as column vectors.
Linear transformations as matrices
Let be vector spaces (over a common field ) of dimension and respectively. Fix bases and for and respectively. We shall order these bases so that and whenever . To distinguish an ordinary set from an ordered set, we shall adopt the notation to mean the set with ordering whenever . The importance of ordering these bases will be apparent shortly.
For any linear transformation , we can write
for each and . We define the matrix associated with the linear transformation and ordered bases by
Let be given by
Using the standard ordered bases
ordered in the obvious way. Then,
so the matrix associated with and the standard ordered bases and is the matrix
Let be the same linear transformation as above. However, let be the same basis as except that the order is reversed: . Then
Note that this matrix is just the matrix from the previous example except that the first and the last columns have been switched.
Again, let be the same as before. Now, let be the ordered basis whose elements are those of but the order is now given by . 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:
the matrix depends on the bases given to the vector spaces
the ordering of a basis is important
switching the order of a given basis amounts to switching columns and rows of the matrix, essentially multiplying a matrix by a permutation matrix.
Some basic properties of matrix representations of linear transformations are
If is a linear transformation, then , where are ordered bases for respectively.
If are linear transformations, then , where and are ordered bases for and respectively.
If and , then , where are ordered bases for respectively.
We could have represented all vectors as row vectors. However, doing so would mean that the matrix representation of a linear transformation would be the transpose of the matrix representation of if the vectors were represented as column vectors: , and that the application of the matrices to vectors would be from the right of the vectors:
Matrices as linear transformations
Every matrix over a field can be thought of as a linear transformation from to if we view each vector as a matrix (a column) and the mapping is done by the matrix multiplication , which is a matrix (a column vector in ). Specifically, we define by
It is easy to see that is indeed a linear transformation. Furthermore, , since the representation of vectors as -tuples of elements in is the same as expressing each vector under the standard basis (ordered) in the vector space . Below we list some of the basic properties:
, for any ,
, where are matrices over
, where is an matrix and is an matrix over
is invertible iff is an invertible matrix.
Remark. As we can see from the discussion above, if we fix sets of base elements for a vector space and , there is a one-to-one correspondence between the set of matrices (of the same size) over the underlying field and the set of linear transformations from to .
- 1 Friedberg, Insell, Spence. Linear Algebra. Prentice-Hall Inc., 1997.
|Title||matrix representation of a linear transformation|
|Date of creation||2013-03-22 17:29:59|
|Last modified on||2013-03-22 17:29:59|
|Last modified by||CWoo (3771)|
|Synonym||standard ordered bases|
|Defines||standard ordered basis|