LA53 What is the Kernel of a Linear Transformation?

Definition, Examples, and Elementary properties of Kernels of Linear Transformations. It is proved that a linear transformation is 1-1 if and only if its kernel consists of only the zero vector. More generally, it is proved that the set of vectors in the domain that are sent by a linear transformation to a specific element in the codomain are translates (fibers, cosets) of the kernel. Subscribe @Shahriari for Math Videos.
00:00 Introduction
04:57 Definition: Kernel of a Linear Transformation
07:06 Theorem: Kernel is a Subspace
07:55 Definition: Nullity = dimension of the kernel
08:12 Proof: Kernel is a subspace
12:41 Example: Finding the Kernel of a LT T : R^3 to R^2
18:08 Theorem: The LT T is 1-1 if and only if ker(T) = {0_V}
19:54 Proof: If ker(T) is just the zero vector then the LT T is 1-1
23:03 Discussion: Utility of the Kernel
24:45 Theorem: A translate of the kernel consists of exactly those elements of the domain that are mapped to a single element in the codomain
27:20 Visual Representation of a Kernel and its Translates (aka cosets or fibers)
30:09 Discussion of the Proof. What needs to be proved?
30:39 Proof: The "easier" direction
32:13 Proof: The other direction
34:50 Application: How are solutions to AX = b related to solutions to AX = 0?
40:14 Application: In Differential Equations, how are solutions to a non-homogeneous linear DE related to solutions of the corresponding homogeneous DE?

This is a video in a series of lectures on linear algebra. The series is a rigorous treatment meant for students with no prior exposure to linear algebra. In this full undergraduate course in linear algebra, general vector spaces and linear transformations are emphasized.

Shahriar Shahriari is the William Polk Russell Professor of Mathematics at Pomona College in Claremont, CA USA

[Shahriari]

Correction. At 18:35 I misspoke twice. A map is onto if everything in the codomain (and not the image) is hit. Everything in the image of the map is automatically hit.