University of Oxford mathematician Dr Tom Crawford introduces the concept of a Linear Transformation with a motivation for the definition and several worked examples. Check out ProPrep with a 30-day free trial: https://www.proprep.uk/info/TOM-Crawford
Links to the other videos mentioned:
Vector Space Axioms - 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/linear-transformation-definition/vid10223
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 formal mathematical definition of a linear transformation and how their properties relate to those of a vector space. There is also a discussion as to how linear maps ensure that the structure of a vector space is preserved.
An alternative definition for a linear transformation is then introduced, which is often used to check whether or not a given map is indeed linear. We also see how a linear transformation must always map the zero vector in the starting space to the zero vector of the final space. A generalised application to a basis of a vector space is also briefly mentioned (this will be discussed further in the next video on the Rank-Nullity Theorem.)
Several examples of linear maps are looked at in detail, including multiplication by a matrix, differentiation, and three explicit maps in R3. The final example re-introduces the concept of a direct sum (as seen in the previous video) and how it can be used to define a projection map.
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
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
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