Linear Algebra 6.2 The Kernel and Range of a Linear Transformation

In this video, we break down four essential ideas in linear algebra: the kernel, range, one‑to‑one transformations, and onto transformations. Whether you’re learning this for the first time or reviewing for a test, this lesson gives you clear explanations, visuals, and examples to help the concepts stick.
You’ll Learn:
– What the kernel of a linear transformation is and how it relates to vectors that map to zero
– How to identify the range (image) of a transformation and understand what outputs are possible
– The meaning of a one‑to‑one (injective) transformation and how the kernel reveals whether a transformation is injective
– What it means for a transformation to be onto (surjective) and how to determine if the entire codomain is covered
– How kernel and range connect to the ideas of dimension, rank, and the behavior of linear maps

Video Chapters:
0:00 Intro
0:08 The Kernel of a Linear Transformation
1:59 Find a Basis for ker(T) as a subspace of R3
5:23 Find a Basis for ker(T) as a subspace of R5 (on your own)
10:02 The Range of a Linear Transformation
11:15 Find a Basis for the Range of T
14:09 Find a Basis for the Range of T (on your own)
15:37 One-to-One Linear Transformations
17:00 One-to-One Example and Non-example
18:30 Onto Linear Transformations
20:24 Summarizing What We Learned
25:31 Isomorphisms and Vector Spaces
26:57 Up Next

Textbook: Elementary Linear Algebra 8e, Ron Larson

Slide Decks: https://bellevueuniversity-my.sharepoint.com/:f:/g/personal/kbrehm_bellevue_edu/IgDVnTt2BohsRLsMfseN5FijAXYDd49YlIkHeq5cfmCxG2g?e=ZwTt3c

Playlist: https://youtube.com/playlist?list=PLl-gb0E4MII1RjTV2GegmXFTJwTrek8OI&si=g3Xc3yHB5GkmxMF4