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: 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 For more maths content check out Tom's website https://tomrocksmaths.com/ You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths. https://www.facebook.com/tomrocksmaths/ https://twitter.com/tomrocksmaths https://www.instagram.com/tomrocksmaths/ Get your Tom Rocks Maths merchandise here: https://beautifulequations.net/collections/tom-rocks-maths