Linear Transformations
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1 a Calculus III Summer 2013, Session II Tuesday, July 23, 2013
2 Agenda a 1. Linear transformations a linear transformation linear transformations
3 a In the m n linear system Ax = 0, Motivation we can regard A as transforming elements of R n (as column vectors) into elements of R m via the rule T (x) = Ax. Then solving the system amounts to finding all of the vectors x R n such that T (x) = 0. Solving the differential equation y + y = 0 is equivalent to finding functions y such that T (y) = 0, where T is defined as T (y) = y + y.
4 a Definition Definition Let V and W be vector s with the same scalars. A mapping T : V W is called a linear transformation from V to W if it satisfies 1. T (u + v) = T (u) + T (v) and 2. T (c v) = c T (v) for all vectors u, v V and all scalars c. V is called the domain and W the codomain of T. Examples T : R n R m defined by T (x) = Ax, where A is an m n matrix T : C k (I) C k 2 (I) defined by T (y) = y + y T : M m n (R) M n m (R) defined by T (A) = A T T : P 1 P 2 defined by T (a + bx) = (a + 2b) + 3ax + 4bx 2
5 a Examples 1. Verify that T : M m n (R) M n m (R), where T (A) = A T, is a linear transformation. The transpose of an m n matrix is an n m matrix. If A, B Mm n (R), then T (A + B) = (A + B) T = A T + B T = T (A) + T (B). If A Mm n (R) and c R, then T (ca) = (ca) T = ca T = c T (A). 2. Verify that T : C k (I) C k 2 (I), where T (y) = y + y, is a linear transformation. If y C k (I) then T (y) = y + y C k 2 (I). If y1, y 2 C k (I), then T (y 1 + y 2 ) = (y 1 + y 2 ) + (y 1 + y 2 ) = y 1 + y 2 + y 1 + y 2 = (y 1 + y 1 ) + (y 2 + y 2 ) = T (y 1 ) + T (y 2 ). If y C k (I) and c R, then T (cy) = (cy) + (cy) = cy + cy = c(y + y) = c T (y).
6 Specifying linear transformations a A consequence of the properties of a linear transformation is that they preserve linear combinations, in the sense that T (c 1 v c n v n ) = c 1 T (v 1 ) + + c n T (v n ). In particular, if {v 1,..., v n } is a basis for the domain of T, then knowing T (v 1 ),..., T (v n ) is enough to determine T everywhere.
7 Linear transformations from R n to R m a Let A be an m n matrix with real entries and define T : R n R m by T (x) = Ax. Verify that T is a linear transformation. If x is an n 1 column vector then Ax is an m 1 column vector. T (x + y) = A(x + y) = Ax + Ay = T (x) + T (y) T (cx) = A(cx) = cax = c T (x) Such a transformation is called a matrix transformation. In fact, every linear transformation from R n to R m is a matrix transformation.
8 a Matrix transformations Theorem Let T : R n R m be a linear transformation. Then T is described by the matrix transformation T (x) = Ax, where A = [ T (e 1 ) T (e 2 ) T (e n ) ] and e 1, e 2,..., e n denote the standard basis vectors for R n. This A is called the matrix of T. Example Determine the matrix of the linear transformation T : R 4 R 3 defined by T (x 1, x 2, x 3, x 4 ) = (2x 1 + 3x 2 + x 4, 5x 1 + 9x 3 x 4, 4x 1 + 2x 2 x 3 + 7x 4 ).
9 a Kernel Definition Suppose T : V W is a linear transformation. The set consisting of all the vectors v V such that T (v) = 0 is called the kernel of T. It is denoted Ker(T ) = {v V : T (v) = 0}. Example Let T : C k (I) C k 2 (I) be the linear transformation T (y) = y + y. Its kernel is spanned by {cos x, sin x}. Remarks The kernel of a linear transformation is a sub of its domain. The kernel of a matrix transformation is simply the null of the matrix.
10 a Definition The range of the linear transformation T : V W is the subset of W consisting of everything hit by T. In symbols, Rng(T ) = {T (v) W : v V }. Example Consider the linear transformation T : M n (R) M n (R) defined by T (A) = A + A T. The range of T is the sub of symmetric n n matrices. Remarks The range of a linear transformation is a sub of its codomain. The range of a matrix transformation is the column of the matrix.
11 a Rank-Nullity revisited Suppose T is the matrix transformation with m n matrix A. We know Hence, Ker(T ) = null(a), dim (Ker(T )) = nullity(a), Rng(T ) = col(a), dim (Rng(T )) = rank(a), the domain of T is R n. dim (domain of T ) = n. We know from the rank-nullity theorem that 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 ).
12 The function of bases a Theorem Let V be a vector with basis {v 1, v 2,..., v n }. Then every vector v V can be written in a unique way as a linear combination v = c 1 v 1 + c 2 v c n v n. In other words, picking a basis for a vector allows us to give coordinates for points. This will allow us to give matrices for linear transformations of vector s besides R n.
13 a linear transformation a Definition Let V and W be vector s with ordered bases B = {v 1, v 2,..., v n } and C = {w 1, w 2,..., w m }, respectively, and let T : V W be a linear transformation. The matrix representation of T relative to the bases B and C is A = [a ij ] where T (v j ) = a 1j w 1 + a 2j w a mj w m. In other words, A is the matrix whose j-th column is T (v j ), expressed in coordinates using {w 1,..., w m }.
14 a Let T : P 1 P 2 be the linear transformation defined by T (a + bx) = (2a 3b) + (b 5a)x + (a + b)x 2. Example Use bases {1, x} for P 1 and {1, x, x 2 } for P 2 to give a matrix representation of T. We have T (1) = 2 5x + x 2 and T (x) = 3 + x + x 2, so 2 3 A 1 = Now use the bases {1, x + 5} for P 1 and {1, 1 + x, x 2 } for P 2. A 1 = 5 1 A 2 = 5 24 We have
15 linear transformations a Definition Let T 1 : U V and T 2 : V W be linear transformations. Their composition is the linear transformation T 2 T 1 defined by (T 2 T 1 ) (u) = T 2 (T 1 (u)). Theorem Let T 1 and T 2 be as above, and let B, C, and D be ordered bases for U, V, and W, respectively. If A 1 is the matrix representation for T 1 relative to B and C, A 2 is the matrix representation for T 2 relative to C and D, A 21 is the matrix representation for T 2 T 1 relative to B and D, then A 21 = A 2 A 1.
16 The inverse of a linear transformation a Definition If T : V W is a linear transformation, its inverse (if it exists) is a linear transformation T 1 : W V such that ( T 1 T ) ( (v) = v and T T 1 ) (w) = w for all v V and w W. Theorem Let T be as above and let A be the matrix representation of T relative to bases B and C for V and W, respectively. T has an inverse transformation if and only if A is invertible and, if so, T 1 is the linear transformation with matrix A 1 relative to C and B.
17 a Let T : P 2 P 2 be defined by Example T (a + bx + cx 2 ) = (3a b + c) + (a c)x + (4b + c)x 2. Using the basis {1, x, x 2 } for P 2, the matrix representation for T is A = This matrix is invertible and A 1 = Thus, T 1 is given by T 1 (a + bx + cx 2 ) = 4a+5b+c 17 + a+3b+4c 17 x + 4a 12b+c 17 x 2.
18 a Theorem Let T : V W be a linear transformation and A be a matrix representation of T relative to some bases for V and W. Ker(T ) = {c 1 v c n v n V : (c 1,..., c n ) null(a)}, Rng(T ) = {c 1 w c m w m W : (c 1,..., c m ) col(a)}.
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