The Kernel and Range Example 3

0:00 T: R^3 to R^2. T(x) = AX.

0:46 Find Ker(T). 8:35 Find Rang (T).

6:35 Give a geometrical description of the Kernel.

9:52 Give a geometrical description of the Range.

6:56 Find dim [Ker(T)]. 10:19 Find dim [Rang(T)].

12:52 Verify the General Rank-Nullity Theorem.

The Kernel Example 4

This is my 5th Video on the Kernel. The Kernel covers more than the topic " Kernel", it covers the Range of the Transformation too. It also covers the Dimension, the basis of the Kernel and the Range, and the use of the General Rank-Nullity Theorem.

I tried to cover the material in less than 15 minutes with notes written before recording to save you some time with clean notes.

0:00 Introduction

1:05 The Kernel

5:26 The Range

9:55 The General Rank-Nullity Theorem

10:18 dim [Pn(R)] = n+1

6.3 The Kernel and Range of a Linear Transformation with Examples.

Here is another video about the Kernel that I created, it is called "The Kernel-Which One Belongs?" https://youtu.be/OYS7qdaChlg

0:00 The Kernel and Range of a Linear Transformation.

0:13 Definition.

1:42 Matrix Transformation: T: R^n to R^m The Kernel is the solution set to the homogenous Linear System AX = 0.

3:06 Ker (T) = nullspace(A).

3:27 Rng(A) = Colspace(A).

 3:57 The General Rank-Nullity Theorem. dim[Ker(T) ] +dim[Rng(T) ]=dim[V. ]

4:41 Example on Ker(T) & Rng(T): Let T: R^3 to R^2 be an L.T. with matrix A [1 -2 5.....]

12:25 Example on Ker(T) & Rng(T): Let S: M2(R) to M2(R) be a L.T. such that S(A) = A - A^T.

19:57 Example on Ker(T) & Rng(T): " be careful here when you Let T: P1 to P2 be a L.T. such that T(a+bx) = (2a-3b) +(b-5a)x+(a+b)x^2.

23:06 Rng(T)=? and what is dim[Rng(T)]. Be careful here. The Kernel and Range " Image" of a Linear Transformation with Examples. The NullSpaces and the ColSpace. The General Rank-Nullity Theorem and Basis. Matrix Transformation. Differential Equations and Linear Algebra.

The Kernel. Which One Belongs?

The Kernel and Range of a Linear Transformation.

Here is the link to the graph in GeoGebra: https://www.geogebra.org/3d/eme2uw29

The Kernel and Range.

Find Ker (T) and Rag (T) for the Linear Transformation T: R^3 to R^3 with the given matrix of transformation A.

Also, give a geometrical description of each.

And find dim[Ker(T)] and dim[Rng(T)] and verify the General Rank-Nullity Theorem.