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Linear Transformations the Matrix of a Linear Trans Linear Trans- formations Math 240 Linear Trans- formations Transformations of Euclidean space Kernel and Range Linear Transformations The matrix of a linear trans. Composition of linear trans. Kernel and Math 240 | Calculus III Range Summer 2013, Session II Tuesday, July 23, 2013 Linear Trans- formations Agenda Math 240 Linear Trans- formations Transformations of Euclidean space Kernel and 1. Linear Transformations Range Linear transformations of Euclidean space The matrix of a linear trans. Composition of linear trans. Kernel and Range 2. Kernel and Range 3. The matrix of a linear transformation Composition of linear transformations Kernel and Range Linear Trans- formations Motivation Math 240 In the m × n linear system Linear Trans- formations Transformations Ax = 0; of Euclidean space n Kernel and we can regard A as transforming elements of R (as column Range vectors) into elements of m via the rule The matrix of R a linear trans. Composition of T (x) = Ax: linear trans. Kernel and Range Then solving the system amounts to finding all of the vectors n x 2 R such that T (x) = 0. Solving the differential equation y00 + y = 0 is equivalent to finding functions y such that T (y) = 0, where T is defined as T (y) = y00 + y: Linear Trans- formations Definition Math 240 Linear Trans- Definition formations Transformations Let V and W be vector spaces with the same scalars. A of Euclidean space mapping T : V ! W is called a linear transformation from Kernel and V to W if it satisfies Range The matrix of 1. T (u + v) = T (u) + T (v) and a linear trans. Composition of 2. T (c v) = c T (v) linear trans. Kernel and Range for all vectors u; v 2 V and all scalars c. V is called the domain and W the codomain of T . Examples n m I T : R ! R defined by T (x) = Ax, where A is an m × n matrix k k−2 00 I T : C (I) ! C (I) defined by T (y) = y + y T I T : Mm×n(R) ! Mn×m(R) defined by T (A) = A 2 I T : P1 ! P2 defined by T (a + bx) = (a + 2b) + 3ax + 4bx Linear Trans- formations Examples Math 240 1. Verify that T : Mm×n(R) ! Mn×m(R), where T Linear Trans- T (A) = A , is a linear transformation. formations Transformations I The transpose of an m × n matrix is an n × m matrix. of Euclidean space I If A; B 2 Mm×n(R), then Kernel and T T T Range T (A + B) = (A + B) = A + B = T (A) + T (B): The matrix of I If A 2 M ( ) and c 2 , then a linear trans. m×n R R Composition of T T linear trans. T (cA) = (cA) = cA = c T (A): Kernel and Range 2. Verify that T : Ck(I) ! Ck−2(I), where T (y) = y00 + y, is a linear transformation. k 00 k−2 I If y 2 C (I) then T (y) = y + y 2 C (I). k I If y1; y2 2 C (I), then 00 00 00 T (y1 + y2) = (y1 + y2) + (y1 + y2) = y1 + y2 + y1 + y2 00 00 = (y1 + y1) + (y2 + y2) = T (y1) + T (y2): k I If y 2 C (I) and c 2 R, then T (cy) = (cy)00 + (cy) = cy00 + cy = c(y00 + y) = c T (y): Linear Trans- formations Specifying linear transformations Math 240 Linear Trans- formations Transformations of Euclidean space Kernel and Range A consequence of the properties of a linear transformation is The matrix of a linear trans. that they preserve linear combinations, in the sense that Composition of linear trans. Kernel and Range T (c1v1 + ··· + cnvn) = c1 T (v1) + ··· + cn T (vn): In particular, if fv1;:::; vng is a basis for the domain of T , then knowing T (v1);:::;T (vn) is enough to determine T everywhere. Linear Trans- n m formations Linear transformations from R to R Math 240 Linear Trans- formations Transformations of Euclidean Let A be an m × n matrix with real entries and define space n m Kernel and T : R ! R by T (x) = Ax. Verify that T is a linear Range transformation. The matrix of a linear trans. Composition of linear trans. I If x is an n × 1 column vector then Ax is an m × 1 Kernel and Range column vector. I T (x + y) = A(x + y) = Ax + Ay = T (x) + T (y) I T (cx) = A(cx) = cAx = c T (x) Such a transformation is called a matrix transformation. In n m fact, every linear transformation from R to R is a matrix transformation. Linear Trans- formations Matrix transformations Math 240 Linear Trans- formations Theorem Transformations n m of Euclidean Let T : ! be a linear transformation. Then T is space R R Kernel and described by the matrix transformation T (x) = Ax, where Range The matrix of A = T (e1) T (e2) ··· T (en) a linear trans. Composition of n linear trans. and e1; e2;:::; en denote the standard basis vectors for . Kernel and R Range This A is called the matrix of T . Example 4 3 Determine the matrix of the linear transformation T : R ! R defined by T (x1; x2; x3; x4) = (2x1 + 3x2 + x4; 5x1 + 9x3 − x4; 4x1 + 2x2 − x3 + 7x4): Linear Trans- formations Kernel Math 240 Linear Trans- Definition formations Transformations Suppose T : V ! W is a linear transformation. The set of Euclidean space consisting of all the vectors v 2 V such that T (v) = 0 is called Kernel and Range the kernel of T . It is denoted The matrix of a linear trans. Ker(T ) = fv 2 V : T (v) = 0g: Composition of linear trans. Kernel and Range Example Let T : Ck(I) ! Ck−2(I) be the linear transformation T (y) = y00 + y. Its kernel is spanned by fcos x; sin xg. Remarks I The kernel of a linear transformation is a subspace of its domain. I The kernel of a matrix transformation is simply the null space of the matrix. Linear Trans- formations Range Math 240 Linear Trans- Definition formations Transformations The range of the linear transformation T : V ! W is the of Euclidean space subset of W consisting of everything \hit by" T . In symbols, Kernel and Range Rng(T ) = fT (v) 2 W : v 2 V g: The matrix of a linear trans. Composition of linear trans. Kernel and Example Range Consider the linear transformation T : Mn(R) ! Mn(R) defined by T (A) = A + AT . The range of T is the subspace of symmetric n × n matrices. Remarks I The range of a linear transformation is a subspace of its codomain. I The range of a matrix transformation is the column space of the matrix. Linear Trans- formations Rank-Nullity revisited Math 240 Linear Trans- Suppose T is the matrix transformation with m × n matrix A. formations Transformations We know Hence, of Euclidean space I Ker(T ) = nullspace(A), I dim (Ker(T )) = nullity(A), Kernel and Range I Rng(T ) = colspace(A), I dim (Rng(T )) = rank(A), The matrix of n a linear trans. I the domain of T is R . I dim (domain of T ) = n. Composition of linear trans. Kernel and We know from the rank-nullity theorem that Range rank(A) + nullity(A) = n: This fact is also true when T is not a matrix transformation: Theorem If T : V ! W is a linear transformation and V is finite-dimensional, then dim (Ker(T )) + dim (Rng(T )) = dim(V ): Linear Trans- formations The function of bases Math 240 Linear Trans- formations Transformations of Euclidean space Kernel and Range Theorem The matrix of Let V be a vector space with basis fv1; v2;:::; vng. Then a linear trans. every vector v 2 V can be written in a unique way as a linear Composition of linear trans. Kernel and combination Range v = c1v1 + c2v2 + ··· + cnvn: In other words, picking a basis for a vector space allows us to give coordinates for points. This will allow us to give matrices n for linear transformations of vector spaces besides R . Linear Trans- formations The matrix of a linear transformation Math 240 Linear Trans- formations Transformations of Euclidean space Definition Kernel and Let V and W be vector spaces with ordered bases Range The matrix of B = fv1; v2;:::; vng and C = fw1; w2;:::; wmg, a linear trans. respectively, and let T : V ! W be a linear transformation. Composition of linear trans. Kernel and The matrix representation of T relative to the bases B Range and C is A = [aij] where T (vj) = a1jw1 + a2jw2 + ··· + amjwm: In other words, A is the matrix whose j-th column is T (vj), expressed in coordinates using fw1;:::; wmg. Linear Trans- formations Example Math 240 Let T : P1 ! P2 be the linear transformation defined by Linear Trans- formations 2 Transformations T (a + bx) = (2a − 3b) + (b − 5a)x + (a + b)x : of Euclidean space 2 Kernel and Use bases f1; xg for P1 and f1; x; x g for P2 to give a matrix Range representation of T . The matrix of a linear trans. Composition of linear trans. We have Kernel and Range T (1) = 2 − 5x + x2 and T (x) = −3 + x + x2; so 2 2 −33 A1 = 4−5 15 : 1 1 2 Now use the bases2 f21; x−+33 5g for P1 and2 f16; 1 + 25x;31 + x g for P2. A1 = 4−5 15 A2 = 4−5 −245 We have 1 1 1 6 T (1) = 2 − 5x + x2 = 6(1) − 5(1 + x) + (1 + x2) and T (x + 5) = 7 − 24x + 6x2 = 25(1) − 24(1 + x) + 6(1 + x2); so Linear Trans- formations Composition of linear transformations Math 240 Linear Trans- formations Definition Transformations of Euclidean space Let T1 : U ! V and T2 : V ! W be linear transformations.
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