Tom Rocks Maths | Oxford Linear Algebra: Rank Nullity Theorem @TomRocksMaths | Uploaded 1 year ago | Updated 7 minutes ago
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: proprep.uk/info/TOM-Crawford
Links to the other videos mentioned:
Linear Transformations: youtu.be/UF59Mok4fsQ
Dimension Formula: youtu.be/lGo9E5dZ4f8
Spanning, Basis & Linear Independence: youtu.be/wdOFi8aUNp0
Subspace Test: 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: 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: 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): youtu.be/9pF__coVyEE
Calculating the inverse of 2x2, 3x3 and 4x4 matrices: youtu.be/VKOaG3Ogf9Q
What is the Determinant Function: youtube.com/watch?v=bLsBWVYSg0A
The Easiest Method to Calculate Determinants: youtu.be/qniUv4EZB0w
Eigenvalues and Eigenvectors Explained: youtu.be/8uISh6xyW7w
Spectral Theorem Proof: youtu.be/ADwsk9G5s_8
Vector Space Axioms: youtu.be/draqOOUoWQM
Subspace Test: youtu.be/3_MxBlWQsgs
Basis, Spanning and Linear Independence: youtu.be/wdOFi8aUNp0
Dimension Formula: youtu.be/lGo9E5dZ4f8
Direct Sum: youtube.com/watch?v=b1VZm4Oecq4
Linear Transformations: 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: seh.ox.ac.uk/people/tom-crawford
For more maths content check out Tom's website tomrocksmaths.com
You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.
facebook.com/tomrocksmaths
twitter.com/tomrocksmaths
instagram.com/tomrocksmaths
Get your Tom Rocks Maths merchandise here:
beautifulequations.net/collections/tom-rocks-maths
Check out Proprep with a 30-day free trial here: proprep.uk/info/TOM-Crawford
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: proprep.uk/info/TOM-Crawford
Links to the other videos mentioned:
Linear Transformations: youtu.be/UF59Mok4fsQ
Dimension Formula: youtu.be/lGo9E5dZ4f8
Spanning, Basis & Linear Independence: youtu.be/wdOFi8aUNp0
Subspace Test: 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: 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: 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): youtu.be/9pF__coVyEE
Calculating the inverse of 2x2, 3x3 and 4x4 matrices: youtu.be/VKOaG3Ogf9Q
What is the Determinant Function: youtube.com/watch?v=bLsBWVYSg0A
The Easiest Method to Calculate Determinants: youtu.be/qniUv4EZB0w
Eigenvalues and Eigenvectors Explained: youtu.be/8uISh6xyW7w
Spectral Theorem Proof: youtu.be/ADwsk9G5s_8
Vector Space Axioms: youtu.be/draqOOUoWQM
Subspace Test: youtu.be/3_MxBlWQsgs
Basis, Spanning and Linear Independence: youtu.be/wdOFi8aUNp0
Dimension Formula: youtu.be/lGo9E5dZ4f8
Direct Sum: youtube.com/watch?v=b1VZm4Oecq4
Linear Transformations: 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: seh.ox.ac.uk/people/tom-crawford
For more maths content check out Tom's website tomrocksmaths.com
You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.
facebook.com/tomrocksmaths
twitter.com/tomrocksmaths
instagram.com/tomrocksmaths
Get your Tom Rocks Maths merchandise here:
beautifulequations.net/collections/tom-rocks-maths
Check out Proprep with a 30-day free trial here: proprep.uk/info/TOM-Crawford
