Linear Transformations on Rn Definition of a Linear Transformation In your travels throughout your mathematical career there has been one theme that persists in every course.  That theme is functions.  Recall that a function is a rule that assigns every element from a domain set to a unique element of a range set.  If the domain and range are both the real numbers, then a function is the familiar real valued function.  If the domain is a real number and the range is Rn then the function is a vector valued function or a parametrically defined curve.  If the domain is Rn and the range is the real numbers, then the function is a function of several variables.  In linear algebra we are interested in special functions where the domain is Rn and the range is Rm.    Definition Let         L:  Rn  --->  Rm be a function such that the following two properties hold: L(u + v)  =  L(u) + L(v) L(cu)  =  cL(u) Then the function is called a linear transformation.   This definition calls for some examples.    Example Let         L:  R2  --->  R3 be defined by         L(x,y)  =  (y, x, x + y) Show that L is a linear transformation.   Solution First, we prove the first property.  Suppose that         u  =  (u1,u2)        and       v  =  (v1,v2) then         L(u + v)  =  L(u1 + v1, u2 + v2)         =  (u2 + v2, u1 + v1, u1 + v1 + u2 + v2) and         L(u) + L(v)  =  (u2, u1, u1 + u2) + (v2, v1, v1 + v2)          =  (u2 + v2, u1 + v1, u1 + u2 +v1 + v2) Hence         L(u + v)  =  L(u) + L(v) Now for the second property.  We have         L(cu)  =  L(c(u1, u2))  =  L(cu1, cu2)          =  (cu2, cu1, cu1 + cu2)  =  c(u2, u1, u1 + u2)  =  cL(u) Since properties 1 and 2 hold, we can conclude that L is a linear transformation. Example Show that the function         f:  R3 --->  R2 defined by         f(x, y, z)  =  (xy, yz) is not a linear transformation   Solution To show a function is not a linear transformation, we just need to find an example that demonstrates the failure of one of the properties.  We have         f(2(3, 4, 5))  =  f(6, 8, 10)  =  (48, 80) and         2(f(3, 4, 5))  =  2(12, 20)  =  (24, 40) since these are not equal, we can conclude that f is not a linear transformation.   It is a simple consequence to the two properties that if L is a linear transformation then         L(c1v1 + c2v2 + ... + ckvk)  =  c1L(v1) + c2L(v2) + ... + ckL(vk) or in sigma notation         L(Scivi)  =  S L(civi) The Matrix of a Linear Transformation   Example/Theorem Let A be an m x n matrix and let         L:  Rn  --->  Rm be defined by         L(u)  =  AuT Then L is a linear transformation.   Proof The proof is just a matter of stating the corresponding properties of matrices. We have         L(u + v)  =  A(u + v)T  =  A(uT + vT)                          =  AuT + AvT  =  L(u) + L(v) and          L(cu)  =  A(cu)T  =  A(cuT)                      =  cAuT  =  cL(u)    The converse is also true.   Theorem Let          L:  Rm  --->  Rn  be a linear transformation.  Then there is a unique matrix A such that          L(u)  =  AuT   Proof Recall that the vector ei is the vector with ith component equal to 1 and all others zero.  Any vector          v  =  (v1,v2, ... ,vm)  =  v1e1 + v2e2 + ... + vmem  We let the ith column of A be the vector         L(ei) Notice that          Aei  =  the ith column of A  so that         L(ei)  =  Aei  Then          L(v)  =  L(v1, v2, ... , vm)  =  L(v1e1 + v2e2 + ... + vmem)           =   v1L(e1) + v2L(e2) + ... + vmL(em)  =  v1Ae1 + v2Ae2 + ... + vmAem          =  A(v1e1 + v2e2 + ... + vmem)  =   AvT  To show that the matrix is unique, we just notice that if B is a matrix with          L(v)  =  BvT then          Bei  =  L(ei)  =  Aei  so that the columns of A and B are the same.   Example For the linear transformation from the first example,         L(x, y)  =  (y, x, x + y) We have         L(1, 0)  =  (0, 1, 1)        L(0, 1)  =  (1, 0, 1) so that the matrix A that represents the linear transformation is         Properties of Linear Transformations There are a few notable properties of linear transformation that are especially useful.  They are the following. L(0)  =  0 L(u - v)  =  L(u) - L(v) Notice that in the first property, the 0's on the left and right hand side are different.  The left hand 0 is the zero vector in Rm and the right hand 0 is the zero vector in Rn.  The proofs of these can be done in two ways.  One way is to use the definition of a linear transformation.  A quicker way is to note that a linear transformation can be represented as a matrix.  These two properties are just properties of matrices.   Back to the Linear Algebra Home Page