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:...
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/downlo...
You can also find several video lectures from ProPrep explaining the content covered in the video here: 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 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:...
Calculating the inverse of 2x2, 3x3 and 4x4 matrices: • Oxford Linear Algebra:...
What is the Determinant Function: • Oxford Linear Algebra:...
The Easiest Method to Calculate Determinants: • Oxford Linear Algebra:...
Eigenvalues and Eigenvectors Explained: • Oxford Linear Algebra:...
Spectral Theorem Proof: • Oxford Linear Algebra:...
Vector Space Axioms: • Oxford Linear Algebra:...
Subspace Test: • Oxford Linear Algebra:...
Basis, Spanning and Linear Independence: • Oxford Linear Algebra:...
Dimension Formula: • Oxford Linear Algebra:...
Direct Sum: • Oxford Linear Algebra:...
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-c...
For more maths content check out Tom's website tomrocksmaths.com/
You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.
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Get your Tom Rocks Maths merchandise here:
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[@Bengolstar300]

Thank you Dr Crawford for giving us this fantastic lesson ❤️

[@Emailminecraft]

I have never discovered the maths side of content on youtube, as far as i could tell yours was the most interesting, i started to binge watch your videos and boy i am happy you are still making videos. Thanks and keeo up the great work ❤❤❤

[@demiladesegun-omosehin7609]

I might actually just stop showing up to class and only watch your videos

[@TomRocksMaths]

Check out ProPrep with a 30-day free trial to see how it can help you to improve your performance in STEM-based subjects: www.proprep.uk/info/TOM-Crawford

[@jeee2025]

I don't like math but watch you give me inspiration to understand math

[@science0064]

Pls make a video on Riemann Hypothesis
Please Dr. Tom

[@learnmathematicswithme847]

How I study topology suggest me a beautiful channel I shall be very thankful

[@darkounet2585]

How the fuck did I end up here. Anyway, seem like a good vid? Can't know for sure i only understood the part where you introduce yourself.

[@cco9162]

Why do you call it Oxford linear algebra? Isn't it the same all over the globe?

[@amritlohia8240]

Yes - it's just to point out that this presentation of it is based on the first-year lecture course taught at Oxford.

[@Tyln93]

Awkward Emo Nerd Boy!!!