Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Signal-flow graph wikipedia , lookup
Tensor operator wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
Matrix calculus wikipedia , lookup
Eigenvalues and eigenvectors wikipedia , lookup
Covariance and contravariance of vectors wikipedia , lookup
Vector space wikipedia , lookup
Four-vector wikipedia , lookup
Cartesian tensor wikipedia , lookup
Basis (linear algebra) wikipedia , lookup
System of linear equations wikipedia , lookup
8.3 Inverse Linear Transformations Definition one-to-one A linear transformation T:V→W is said to be one-to-one if T maps distinct vectors in V into distinct vectors in W . Example 1 A One-to-One Linear Transformation Recall from Theorem 4.3.1 that if A is an n×n matrix and TA :Rn→Rn is multiplication by A , then TA is one-toone if and only if A is an invertible matrix. Example 2 A One-to-One Linear Transformation Let T: Pn → Pn+1 be the linear transformation T (p) = T(p(x)) = xp(x) Discussed in Example 8 of Section 8.1. If p = p(x) = c0 + c1 x +…+ cn xn and q = q(x) = d0 + d1 x +…+ dn xn are distinct polynomials, then they differ in at least one coefficient. Thus, T(p) = c0 x + c1 x2 +…+ cn xn+1 and T(q) = d0 x + d1 x2 +…+ dn xn+1 Also differ in at least one coefficient. Thus, since it maps distinct polynomials p and q into distinct polynomials T (p) and T (q). Example 3 A Transformation That Is Not One-to-One Let D: C1(-∞,∞) → F (-∞,∞) be the differentiation transformation discussed in Example 11 of Section 8.1. This linear transformation is not one-to-one because it maps functions that differ by a constant into the same function. For example, D(x2) = D(xn +1) = 2x Equivalent Statements Theorem 8.3.1 If T:V→W is a linear transformation, then the following are equivalent. (a) (b) (c) T is one-to-one The kernel of T contains only zero vector; that is , ker(T) = {0} Nullity (T) = 0 Theorem 8.3.2 If V is a finite-dimensional vector space and T:V ->V is a linear operator then the following are equivalent. (a)T is one to one (b) ker(T) = {0} (c)nullity(T) = 0 (d)The range of T is V;that is ,R(T) =V Example 5 Let T A:R 4 -> R 4 be multiplication by 1 2 A= 3 1 3 2 4 6 4 8 9 1 5 1 4 8 Determine whether T A is one to one. Example 5(Cont.) Solution: det(A)=0,since the first two rows of A are proportional and consequently A I is not invertible.Thus, T A is not one to one. Inverse Linear Transformations If T :V -> W is a linear transformation, denoted by R (T ),is the subspace of W consisting of all images under T of vector in V. If T is one to one,then each vector w in R(T ) is the image of a unique vector v in V. Inverse Linear Transformations This uniqueness allows us to define a new function,call the inverse of T. denoted by T –1.which maps w back into v(Fig 8.3.1). Inverse Linear Transformations T –1:R (T ) -> V is a linear transformation. Moreover,it follows from the defined of T –1 that T –1(T (v)) = T –1(w) = v T –1(T (w)) = T –1(v) = w (2a) (2b) so that T and T –1,when applied in succession in either the effect of one another. Example 7 Let T :R 3 ->R 3 be the linear operator defined by the formula T (x1,x2,x3)=(3x1+x2,-2x1-4x2+3x3,5x1+4 x2-2x3) Solution: 3 1 0 2 4 3 [T ]= 5 4 2 4 2 3 11 6 9 -1 ,then[T ] = 12 7 10 Example 7(Cont.) T x1 –1 x 2 x3 =[T x1 –1] x 2 = x 3 4 2 3 x1 11 6 9 x 2 12 7 10 x 3 4 x1 2 x 2 3x3 11x1 6 x 2 9 x3 = 12 x1 7 x 2 10 x3 Expressing this result in horizontal notation yields T –1(X1,X2,X3)=(4X1-2X2-3X3,-11X1+6X2+9X3,-12X1+7X2+10X3) Theorem 8.3.3 If T1:U->V and T2:V->W are one to one linear transformation then: (a)T2 0 T1 is one to one (b) (T2 0 T1)-1 = -1 -1 T1 0 T2