6.4 Additional Properties of Linear Transformation.

0:00 Additional Properties of Linear Transformation with Examples.

 0:17 Definition. 2:33 Theorem " Linear Transformation".

2:58 Example: T1 : R^n to R^m and T2: R^m to R^p; T1(x) = AX and T2(x) = BX. Find T2T1(X) and the Matrix of Transformation that takes you from R^n to R^p.

6:05 Example: T1: Mn(R) to Mn (R); T1 (A) = A +A^T. T2: Mn(R) to R; T2( A) + tr(A). What is T2T1(x) =?

9:11 Definition: 1-1 and Onto.

10:23 Example: T M2(R) to R ; T(A) = tr (A). [ T is NOT 1-1], and [ t is Onto]. 12:56 Theorem T: V to W. a) 1-1 iff Kr(T) = {0}. b) Onto iff Rng (T) = W

14:06 Example: T: R^7 to R^5 is a L. T. such that dim[Ker(T)} = 2. Show that T is ONTO.

16:06 Example T: R^3 to R^6 is a L. T. such that dim[Ker(T)} = 3. Show that T is 1-1.

17:45 Example T: P6 to R^5 is a L. T. such that dim[Ker(T)} = 2. Show that T is ONTO.

19:28 Example: T: P2 to P2; T(a+bx+cx^2) = (2a-6+c) + ( b-2a)x +cx^2 Determine whether T is: a) 1-1. b) ONTO. c) Both. d) Neither. The answer is: d) Neither.

 27:30 Theorems.

30:32 Definition: Isomorphism.

31:15 Example on Isomorphism between R^4 and M2(R). Composition, or Product of a Transformation. One-to-one, Onto, and Isomorphism. Differential Equations and Linear Algebra.