Oxford Linear Algebra: Basis, Spanning and Linear Independence

University of Oxford mathematician Dr Tom Crawford explains the terms basis, spanning and linear independence in the context of vectors and vector spaces. 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 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-%20Vector%20Spaces%20over%20R%20(Rn)%20-%20workbook.pdf And 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. Basis for R^n: https://www.proprep.com/courses/all/linear-algebra/vector-spaces-over-r-(rn)/basis-for-rn/vid9962 Basis: https://www.proprep.com/courses/all/linear-algebra/general-vector-spaces/vector-basis/vid25730 Spanning: https://www.proprep.com/courses/all/linear-algebra/general-vector-spaces/linear-combination,-dependence-and-span/vid25722 Linear Independence: https://www.proprep.com/courses/all/linear-algebra/vector-spaces-over-r-(rn)/linear-combination,-dependence-and-span/vid10012 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 The video begins with an intuitive example of a basis via the vector space of polynomials up to degree n. We then give the formal definition of a basis as a spanning set of linearly independent vectors. The terms spanning and linear independence are then formally defined with examples given for each. We also show the definition of linear independence is equivalent to showing that the only solution to a linear combination of the vectors being equal to zero is for all of the coefficients to be zero. Linear dependence is defined as the lack of linear independence, or when a vector in a set can be written as a linear combination of the other vectors in the set. Finally, we move on to a series of worked examples, beginning with several possible bases for the Cartesian plane R^2. We then look at examples of linearly independent and linearly dependent sets of vectors, and how to show this is the case. Finally, we construct two possible bases for 3D space R^3. 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