2 Recall that the difference between the matrix equation and the associated vector equationis just a matter of notation.However the matrix equation can arise is linear algebra (and applications) in a way that is not directly connected with linear combinations of vectors.This happens when we think of a matrix A as an object that acts on a vector by multiplication to produce a new vector
6 In the previous example, solving the equation Ax = b can be thought of as finding all vectors x in R4 that are transformed into the vector b in R2 under the “action” of multiplication by A.
7 Transformation:Any function or mappingTRangeDomainCodomain
8 Let A be an mxn matrix.Matrix Transformation:ACodomainDomainxbA
9 Example: The transformation T is defined by T(x)=Ax where For each of the following determine m and n.
10 Matrix Transformation: A x = bxAbDomainCodomain
11 Linear Transformation: Definition:A transformation T is linear if(i) T(u+v)=T(u)+T(v) for all u, v in the domain of T:(ii) T(cu)=cT(u) for all u and all scalars c.Theorem: If T is a linear transformation, thenT(0)=0 andT(cu+dv)=cT(u)+dT(v) for all u, v and all scalars c, d.
12 Example. Suppose T is a linear transformation from R2 to R2 such that and With no additionalinformation, find a formula for the image of an arbitrary x in R2.
14 Theorem 10.Let be a linear transformation. Then there exists a unique matrix A such that for all x in Rn.In fact, A is the matrix whose jth column is the vector where is the jth column of the identity matrix in Rn.A is the standard matrix for the linear transformation T
15 Find the standard matrix of each of the following transformations. Reflection throughthe x-axisReflection throughthe y-axisReflection throughthe y=xReflection throughthe y=-xReflection throughthe origin
16 Find the standard matrix of each of the following transformations. HorizontalContraction &ExpansionVerticalContraction &ExpansionProjection ontothe x-axisProjection ontothe y-axis
17 Applets for transformations in R2 From Marc Renault’s collection…Transformation of PointsVisualizing Linear Transformations
18 DefinitionA mapping is said to be ontoif each b in is the image of at least one x inDefinitionA mapping is said to be one-to-oneif each b in is the image of at most one x inTheorem 11Let be a linear transformation. Then,T is one-to-one iff has only the trivial solutionTheorem 12Let be a linear transformation with standard matrix A.1. T is onto iff the columns of A span2. T is one-to-one iff the columns of A are linearly independent