Chapter 10 Linear Transformations and Matrices
Definition of linear transformation: n A linear transformation L of R n into R m is a function assigning a unique vector L(u) in R m to each u in R n (denoted by L: V à W ) such that for n
u, v Î R , k Î R (a) L(u + v ) = L (u ) + L (v ) (b) L(ku ) = kL (u ) Examples Example 1(page 503) n Example 2(page 503) n Example 3(page 503) n Example 4(page 504) n Example 5(page 504) n Example 6(page 504) n Example 7(page 504)
n Note: n n If L: V à V and L is a linear transformation, L is also called a linear operator on V. L(u) is called the image of u and the set of all images of the vectors in R m is called the range of L.
Matrix Transformation n Let A be an m*n matrix. We defined a matrix transformation as a function L: R n m à R defined by L(u)= Au. We can easily show that every matrix transformation is a linear transformation. Since
(a) L (u + v ) = A ( u + v ) =
Au + Av = L (u ) + L (v ) (b) L (ku ) = A ( ku ) = k ( Au ) = kL (u ) Case of Matrix Transformation Reflection with respect to the x­axis n Projection into the xy­plane n Dilation n Contraction n Rotation counterclockwise n Shear in the x­direction n Shear in the y­direction
n Theorem 10.1(Theorem 4.6) Let L: V (R n ) àW (R m ) be a linear transformation. Then, n for any vector v 1 , v 2 ,… , v k and any scalars c 1 , c 2 ,… , c k , then
L(c 1v 1 + c 2 v 2 +L + c k v k )
= c 1 L (v 1 ) + c 2 L (v 2 ) +L+ c k L (v k ) Theorem 4.5(extension) If V is an n­dimensional vector space and
S = {w 1 , w 2 , K , w n } be a basis for V. If u is any vector in V, then
L (u ) is a linear combination of
.
L (w 1 ), L (w 2 ), K , L (w n ) Theorem 10.2(Theorem 4.7) Let L: V (R n ) àW (R m ) be a linear transformation. Then, n L(0 v ) à 0 w , where 0 v is the zero vector in R n and 0 w is the zero vector in R m . n L(u­v)=L(u) ­ L(v), for u and v in R n .
Corollary 10.1(Corollary 4.1) Let T: V (R n ) àW (R m ) be a transformation. n If T(0 v ) ≠ 0 w , where 0 v is the zero vector in R n and 0 w is the zero vector in R m . Then T is not a linear transformation.
Theorem 10.3 n Let L:V à W be a linear transformation of an n­dimensional vector space V into a vector space W. Also, let S={v 1, v 2,…,
v n} be a basis for
V. If u is any vector in V, then L(u) is completely determined by { L(v 1),
L(v n)}. L(v 2),….,
Example 8(page 506)
Theorem 4.8 Let L: R n àR m be a linear transformation. Then there exists a unique m*n matrix A such that,
n L ( x ) = Ax , x Î R ,where
A = [L (e 1 ) L (e 2 ) L L (e n )] and
é 1 ù
é 0 ù
é 0 ù
ê 0 ú
ê 1 ú
ê 0 ú
e1 = ê ú , e 2 = ê ú , L , e n = ê ú
ê M ú
êMú
êMú
ê ú
ê ú
ê ú
0 0 ë û
ë û
ë 1 û
Example æ é x 1 ù ö é 2 x 1 + x 2 ù
L : R ® R , L ( x ) = L çç ê ú ÷÷ = ê
. ú
è ë x 2 û ø ë 3 x 2 û
2
n 2 Please determine the matrix such that
L(x ) = Ax , x Î R 2 é2 1 ù é x 1 ù é2 x 1 + x 2 ù
L( x ) = Ax = ê
=ê
ê
ú
ú
ú
ë0 3 û ë x 2 û ë 3 x 2 û
Section 10.2 The Kernel and Range of a Linear Transformation
Definition of one­to­one transformation: n A linear transformation L:V à W is said to be one­to­one if for all v 1, v 2 in V, v 1≠ v 2 implies that L(v 1 )≠ L(v 2 ) (or L(v 1 )= L(v 2 ) implies that v 1= v 2 ).
æ é x ùö éx - y ù
L çç ê ú÷÷ = ê
ú
è ë y ûø ëx + y û
Is L one­to­one?
Let
é x 1 ù
é x 2 ù
v 1 = ê ú , v 2 = ê ú
ë y 1 û
ë y 2 û
. Then,
éx 1 - y 1 ù éx 2 - y 2 ù
L (v 1 ) = L (v 2 ) Ûê
=ê
Û
ú
ú
ëx 1 + y 1 û ëx 2 + y 2 û
x1 - y 1 = x 2 -y 2 x 1 + y 1 = x 2 + y 2 . Û 2 x 1 = 2 x 2 Û x 1 = x 2 Û y 1 = y 2 Û
v
1 =
é x 1 ù
ê y ú =
ë 1 û
é x 2 ù
ê y ú = v 2 2 û
ë
Therefore, L is one­to­one.
Examples Example 1(page 509) n Example 2(page 509)
n Definition of kernel: Let L:V à W be a linear transformation. The kernel of L, Ker(L), is the subset of V consisting of all vectors v, such that L(v)=0 W. What is ker(L)? æ é u 1 ù ö
çê ú÷
L (u ) = L ç ê u 2 ú ÷ = Au =
ç ê u ú ÷
è ë 3 û ø
é 1 2 3 ù é u 1 ù
ê 4 5 6 ú ê u ú
ê
ú ê 2 ú
êë 7 8 9 úû êë u 3 úû
ker(L) is the set consisting of all vectors
such that
é u 1 ù
u = êê u 2 úú
êë u 3 úû
L (u ) = Au = 0
.
That is, ker(L) is the solution space of Au=0. Examples Example 3(page 510) n Example 4(page 510) n Example 5(page 510)
n Theorem 10.4 If L:V à W is a linear transformation, then Ker(L) is a subspace of V. [proof:] For any
v1, v 2 Î ker (L ) . Then, 1.
L (v 1 ) = L (v 2 ) = 0 L(v 1 + v 2 ) = L (v 1 ) + L (v 2 ) = 0 + 0 = 0 Þ v 1 + v 2 Î ker (L ) 2 . L(kv 1 ) = kL (v 1 ) = k 0 = 0 , k Î R Þ kv 1 Î ker (L ) By 1, 2, ker (L) is a subspace of V.
Examples Example 6(page 511) n Example 7(page 511) n Example 8(page 511)
n Theorem 10.5 A linear transformation L:V à W is one­to­one if and only if Ker(L) ={0 v }. [proof:] Þ :
Since L is one­to­one, L(0)=0 implies there is only . one vector 0 such that L(0)=0. That is ker(L)= 0 Ü : Let ker(L)= {0}. Suppose L(u)=L(v). Then,
L (u - v ) = L (u ) - L (v ) = 0 . Since
ker (L ) = {0 } ,that implies u – v = 0 => u = v. Therefore, L is one­to­one.
Important result: Let L:R n à R n be a linear transformation defined by L(x)=Ax , where A is a n*n matrix. Then, the following conditions are equivalent: n Ker(L)= {0}. n Ax=0 has only the trivial solution. n Ax=b has a unique solution for every b. n L(x)=Ax is one­to­one.
Corollary 10.2: n If L(x) = b and L(y) = b, then x­y belongs to ker L. In other words, any two solutions to L(x) =b differ by an element of the kernel of L. The linear transformation in Example 1 is one­ to­one; the one in Example 2 is not.
Definition of range: Let L:V à W be a linear transformation. The range of L, range(L) , is the subset of W consisting of all vectors that are images of vectors in V (i.e., for any w in range(L), there exists v in V such that L(v)=w). n If range(L)= W, L is said to be onto.
Theorem(10.6): n Let L:V à W be a linear transformation. Then, range(L) is a subspace of W. [proof:] Since L(0)=0 , 0 is in range(L). Thus range(L) is not empty. For any w 1 , w 2 in range(L), there exist v 1 , v 2 in V such that L(v 1 )=w 1 , L(v 2 )=w 2 . Then 1. w 1 +w 2 =L(v 1 )+L(v 2 )=L(v 1 +v 2 ), which implies w 1 +w 2 is in range(L) . 2. L(kv 1 )=kL(v 1 )=k w 1 , k is a real number. Then kw 1 is in range(L). By 1, 2, range(L) is a subspace of W.
é 1 0 1 ù é x 1 ù
L ( x ) = Ax = êê 1 1 2 úú êê x 2 úú
êë 2 1 3 úû êë x 3 úû
n n n n 3 3 L : R ® R Example 11 page 512
Is L onto? Find a basis for range L. Find ker L. Is L one­to­one? é 1 0 1 ù é x 1 ù
L ( x ) = Ax = êê 1 1 2 úú êê x 2 úú
êë 2 1 3 úû êë x 3 úû
n 3 3 L : R ® R Example 11 page 512 Is L onto? é1 0 1 ù éa ù é1 0 1 ù é a ù
ê1 1 2 ú êb ú ® ê0 1 1 ú ê b - a ú
ê
úê ú ê
úê
ú
êë2 1 3 úû êë c úû êë0 0 0 úû êëc - b - a úû
Thus a solution exists only when c - b - a = 0 . Hence , L is not onto. é 1 0 1 ù é x 1 ù
L ( x ) = Ax = êê 1 1 2 úú êê x 2 úú
êë 2 1 3 úû êë x 3 úû
3 3 L : R ® R Example 11 page 512 Find a basis for range L. é 1 0 1 ù é a ù é a + c ù
é 1 ù
é 0 ù
é 1 ù
ê 1 1 2 ú ê b ú = ê a + b + 2 c ú = a ê 1 ú + b ê1 ú + c ê 2 ú
ê
úê ú ê
ú
ê ú
ê ú
ê ú
êë 2 1 3 úû êë c úû êë 2 a + b + 3 c úû
êë 2 úû
êë1 úû
êë 3 úû
That is, range L is the subspace of R 3 spanned by the columns of the matrix defining L . Note that col 1 ( A ) + col 2 ( A ) = col 3 ( A ). The third vector is the sum of the first two. dim(range( L)) = 2. n é 1 0 1 ù é x 1 ù
L ( x ) = Ax = êê 1 1 2 úú êê x 2 úú
êë 2 1 3 úû êë x 3 úû
n n é 1 ê 1 ê
êë 2 3 3 L : R ® R Example 11 page 512 Find ker L. Is L one­to­one? 0 1 1 ù
é 1 0 1 ù
2 úú ® êê 0 1 1 úú
êë 0 0 0 úû
1 3 úû
Solving the resulting homogeneou s system, we get é - 1 ù
ker ( L ) = span{ êê - 1 úú }, and dim( ker ( L )) = 1. êë 1 úû
Since ker ( L ) ¹ {0 R 3 }, L is not one ­ to ­ one. Note: n n The problem of finding a basis for ker L always reduces to the problem of finding a basis for the solution space of a homogeneous system. If range L is a subspace of R m , then a basis for range L can be obtained by the method discussed in the alternative constructive proof of Theorem 6.6 or by the procedure given in Section 6.6. Both approaches are illustrated in the next example. Example 12(page 513)
Theorem 10.7: Let L:V à W be a linear transformation of an n­dimensional vector space V into a vector space W, then, n Dim(ker(L))+Dim(range(L))=dim(V) The dimension of ker L is also called the nullity of L, and the dimension of range L is called the rank of L. (similar to that of theorem 6.12)
Note: As the linear transformation defined by
L : R ® R , L ( x ) = A m ´ n x , n
m ker (L ) = the null space of A range (L ) = the column space of A .
Note: As the linear transformation defined by
L : R ® R , L ( x ) = A m ´ n x , n
m rank ( A ) + nullity ( A )
= dim (column space of A ) + dim (null space of A )
= dim (range (L )) + dim (ker (L ))
( ) = dim R n = n .
Corollary 10.3: Let L:V à W be a linear transformation and let dim V = dim W, then, n If L is one­to­one, then it is onto. n If L is onto, then it is one­to­one.
Examples Example 13(page 516) n Example 14(page 517)
n