Oxford Linear Algebra: Dimension Formula for Vector Spaces

University of Oxford mathematician Dr Tom Crawford introduces the dimension formula for vector spaces via a worked example before going through a complete proof. Check out ProPrep with a 30-day free trial to see how it can help you to improve your performance in STEM-based subjects: https://www.proprep.uk/info/TOM-Crawford Links to the other videos mentioned: Subspace Test - https://youtu.be/3_MxBlWQsgs Basis, Spanning and Linear Independence - https://youtu.be/wdOFi8aUNp0 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-%20General%20Vector%20Spaces%20-%20workbook.pdf You can also find several video lectures from ProPrep explaining the content covered in the video at the links below. Dimension Formula: https://www.proprep.com/university-of-warwick/warwick-university/ma106/general-vector-spaces/subspaces/vid30223 Sum of Subspaces: https://www.proprep.com/university-of-warwick/warwick-university/ma106/general-vector-spaces/subspaces/vid30220 As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises. 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 The video begins by defining the dimension of a vector space as the number of elements in its basis. This is then exemplified by looking at the vector space of polynomials up to degree n, which has a dimension of n+1. We then go through a fully worked example in R^4 by calculating explicitly the dimensions of the subspaces X and Y, the subspace X+Y, and the intersection of X and Y. This is used as motivation for the dimension formula: dim(X+Y) = dim(X) + dim(Y) - dim(X and Y). Finally, a complete proof of the dimension formula is presented where we construct a basis of the space X+Y which is shown to be both spanning and linearly independent. 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