Oxford Linear Algebra: Rank Nullity Theorem

University of Oxford mathematician Dr Tom Crawford introduces the concepts of rank and nullity for a linear transformation, before going through a full step-by-step proof of the Rank Nullity Theorem. Check out ProPrep with a 30-day free trial: https://www.proprep.uk/info/TOM-Crawford Links to the other videos mentioned: Linear Transformations: https://youtu.be/UF59Mok4fsQ Dimension Formula: https://youtu.be/lGo9E5dZ4f8 Spanning, Basis & Linear Independence: https://youtu.be/wdOFi8aUNp0 Subspace Test: https://youtu.be/3_MxBlWQsgs Test your understanding of the content covered in the video with some practice exercises courtesy of ProPrep. You can download the workbooks and solutions for free here: https://api.proprep.com/course/downloadbook?file=Proprep%20-%20Linear%20Algebra%20-%20Linear%20Transformations%20-%20workbook.pdf You can also find several video lectures from ProPrep explaining the content covered in the video here: https://www.proprep.com/courses/all/linear-algebra/linear-transformations/image-and-kernel/vid10197 As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises. The video begins with the definition of the kernel and image of a linear transformation, as well as their dimensions as the nullity and rank of the linear map. Next is a fully worked example of a map in R^2 with an explicit calculation of the kernel, image, nullity and rank. In the second part of the video the Rank Nullity Theorem is presented: the dimension of the starting vector space of a linear map is equal to the rank of the linear map (dimension of its image) plus the nullity of the map (dimension of its kernel). A full step-by-step proof is then presented by constructing a basis for each component. Watch the other videos from the Oxford Linear Algebra series at the links below. Solving Systems of Linear Equations using Elementary Row Operations (ERO’s): https://youtu.be/9pF__coVyEE Calculating the inverse of 2x2, 3x3 and 4x4 matrices: https://youtu.be/VKOaG3Ogf9Q What is the Determinant Function: https://www.youtube.com/watch?v=bLsBWVYSg0A The Easiest Method to Calculate Determinants: https://youtu.be/qniUv4EZB0w Eigenvalues and Eigenvectors Explained: https://youtu.be/8uISh6xyW7w Spectral Theorem Proof: https://youtu.be/ADwsk9G5s_8 Vector Space Axioms: https://youtu.be/draqOOUoWQM Subspace Test: https://youtu.be/3_MxBlWQsgs Basis, Spanning and Linear Independence: https://youtu.be/wdOFi8aUNp0 Dimension Formula: https://youtu.be/lGo9E5dZ4f8 Direct Sum: https://www.youtube.com/watch?v=b1VZm4Oecq4 Linear Transformations: https://youtu.be/UF59Mok4fsQ Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: https://www.seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website https://tomrocksmaths.com/ You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths. https://www.facebook.com/tomrocksmaths/ https://twitter.com/tomrocksmaths https://www.instagram.com/tomrocksmaths/ Get your Tom Rocks Maths merchandise here: https://beautifulequations.net/collections/tom-rocks-maths Check out Proprep with a 30-day free trial here: https://www.proprep.uk/info/TOM-Crawford