Kernel and Range of Linear Transformations | Linear Algebra

We introduce the kernel and range of a linear transformation. The kernel is the extension of the null space to general linear transformations and the range is the extension of the column space to general linear transformations. We go over the definitions of these terms, see several examples, and then prove a theorem stating the kernel is a subspace of the domain and the range is a subspace of the codomain. #linearalgebra Join Wrath of Math to get exclusive videos, lecture notes, and more: https://www.youtube.com/channel/UCyEKvaxi8mt9FMc62MHcliw/join Linear Transformations in Vector Spaces: https://youtu.be/ROGG72W4_00 Linear Algebra course: https://www.youtube.com/playlist?list=PLztBpqftvzxWT5z53AxSqkSaWDhAeToDG Linear Algebra exercises: https://www.youtube.com/playlist?list=PLztBpqftvzxVmiiFW7KtPwBpnHNkTVeJc Get the textbook for this course! https://amzn.to/43xAWEz 0:00 Intro 1:54 Definitions of Kernel and Range 4:06 Zero Transformation Kernel and Range 4:59 Kernel and Range of Projection onto xy Plane 6:51 Kernel and Range of Rotation 7:38 Subspace Theorem 8:14 Kernel is a Subspace 10:28 Range is a Subspace 13:20 Conclusion ★DONATE★ ◆ Support Wrath of Math on Patreon: https://www.patreon.com/join/wrathofmathlessons ◆ Donate on PayPal: https://www.paypal.me/wrathofmath Outro music by Ben Watts and is available for channel members. Follow Wrath of Math on... ● Instagram: https://www.instagram.com/wrathofmathedu ● TikTok: https://www.tiktok.com/@wrathofmathedu ● X: https://x.com/wrathofmathedu ● Facebook: https://www.facebook.com/WrathofMath