Answer:
We define linear transformation for abstract vector spaces, and illustrate the definition with examples.
LTR-0022: Linear Transformations of Abstract Vector Spaces
Recall that a transformation is called a linear transformation if the following are true for all vectors and in , and scalars .
We generalize this definition as follows.
Let and be vector spaces. A transformation is called a linear transformation if the
following are true for all vectors and in , and scalars .
Recall that is the set of all matrices. In Example ex:setofmatricesvectorspace of VSP-0050 we demonstrated
that together with operations of matrix addition and scalar multiplication is a vector
space.
Let be a transformation defined by , where is fixed matrix. Show that is a linear transformation.
We verify the linearity properties using properties of matrix-matrix
and matrix-scalar multiplication. (See Theorem th:propertiesofmatrixmultiplication of MAT-0020.) For and in and a
scalar we have:
Recall that is the set of polynomials of degree or less than . In Example ex:pnisavectorspace of
VSP-0050 we showed that together with operations of polynomial addition and
scalar multiplication is a vector space.
Suppose is a linear transformation such that Find the image of under .
Coordinate Vectors
Transformations that map vectors to their coordinate vectors will prove to be of great importance. In this section we will prove that such transformations are linear and give several examples.
Recall that if is a vector space, and is a basis for then any vector of can be written as a unique linear combination of the elements of . In other words, for some scalars . The vector in whose components are the coefficients is said to be the coordinate vector for with respect to and is denoted by . (Definition def:coordvector of VSP-0060.)
It turns out that the transformation defined by is linear. Before we prove linearity of , consider the following examples.
Consider . Let be a basis for . (You should do a quick mental check that is a
legitimate basis.) Define by . Find .
Recall that is the set of polynomials of degree or less than . In Example ex:deg_le_2vectorspace of
VSP-0050 we showed that is a vector space.
- (a)
- It is easy to verify that is a basis for . If is given by , find .
- (b)
- In Practice Problem prob:linindabstractvsp1 of VSP-0060, you demonstrated that is also a basis for . If is given by , find .
item:lintranspolycoordvect1 We express as a linear combination of elements of .
Therefore
Note that it is important to keep the basis elements in the same order in which they
are listed, as the order of components of the coordinate vector depends on the order
of the basis elements. We conclude that
item:lintranspolycoordvect2 Our goal is to express as a linear combination of the elements of . Thus, we need to find coefficients , and such that This gives us a system of linear equations: Solving the system yields , and . Thus
Let be an -dimensional vector space, and let be a basis for . Then given by is a
linear transformation.
- Proof
- First observe that Theorem th:uniquerep of VSP-0060 guarantees that there is only
one way to represent each element of as a linear combination of elements of .
Thus each element of maps to exactly one element of , as long as the order in
which elements of appear is taken into account. (The order of elements of is
important as it determines the order of components of the coordinate vectors.)
This proves that is a function, or a transformation. We will now prove that is
linear.
Let be an element of . We will first show that . Suppose , then can be written as a unique linear combination: We have:
We leave it to the reader to verify that . (See Practice Problem prob:completeproofoflin.)
Practice Problems
This problem requires the knowledge of how to compute a determinant. (For a quick
reminder, see Definition def:threedetcrossprod of VEC-0080.) Define by .
Recall that the set of all symmetric matrices is a subspace of . In Example ex:symmetricmatsubspace of
VSP-0060 we demonstrated that is a basis for . Define by . Find and
.
Answer:
If the order of the basis elements in Problem prob:coordvector was switched to form a new
basis
How would this affect the coordinate vector?
In Practice Problem prob:linindabstractvsp123 of VSP-0060 you demonstrated that is a basis for . Define by .
Find , and .
Answer: