Oxford Linear Algebra: Linear Transformations Explained

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: www.proprep.uk/info/TOM-Crawford

Links to the other videos mentioned:
Vector Space Axioms -    • 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: api.proprep.com/course/downloadbook?file=Proprep%2…

You can also find several video lectures from ProPrep explaining the content covered in the video here: www.proprep.com/courses/all/linear-algebra/linear-…

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):    • 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 ...  

Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: www.seh.ox.ac.uk/people/tom-crawford

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