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:    • Oxford Linear Algebra: Linear Transfo...  
Dimension Formula:    • Oxford Linear Algebra: Dimension Form...  
Spanning, Basis & Linear Independence:    • Oxford Linear Algebra: Basis, Spannin...  
Subspace Test:    • Oxford Linear Algebra: Subspace Test  

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/downlo...

You can also find several video lectures from ProPrep explaining the content covered in the video here: https://www.proprep.com/courses/all/l...

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):    • Oxford Linear Algebra: Elementary Row...  

Calculating the inverse of 2x2, 3x3 and 4x4 matrices:    • Oxford Linear Algebra: How to find a ...  

What is the Determinant Function:    • Oxford Linear Algebra: What is the De...  

The Easiest Method to Calculate Determinants:    • Oxford Linear Algebra: The Easiest Me...  

Eigenvalues and Eigenvectors Explained:    • Oxford Linear Algebra: Eigenvalues an...  

Spectral Theorem Proof:    • Oxford Linear Algebra: Spectral Theor...  

Vector Space Axioms:    • Oxford Linear Algebra: What is a Vect...  

Subspace Test:    • Oxford Linear Algebra: Subspace Test  

Basis, Spanning and Linear Independence:    • Oxford Linear Algebra: Basis, Spannin...  

Dimension Formula:    • Oxford Linear Algebra: Dimension Form...  

Direct Sum:    • Oxford Linear Algebra: Direct Sum of ...  

Linear Transformations:    • Oxford Linear Algebra: Linear Transfo...  

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-c...

For more maths content check out Tom's website https://tomrocksmaths.com/

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Check out Proprep with a 30-day free trial here: https://www.proprep.uk/info/TOM-Crawford