University of Oxford Mathematician Dr Tom Crawford introduces the concept of a Bilinear Form, Inner Product, Sesquilinear Form and Inner Product Space. Check out ProPrep with a 30-day free trial: https://www.proprep.uk/info/TOM-Crawford Links to the other videos mentioned: Linear Transformations: https://www.youtube.com/watch?v=UF59Mok4fsQ Spanning, Basis & Linear Independence: https://www.youtube.com/watch?v=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-%20Inner%20Product%20Spaces%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/inner-product-spaces/inner-product-spaces/vid10133 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 definition of a Bilinear Form with a concrete example of the dot product on R^n. This is shown to also satisfy the criteria to be symmetric and positive definite, thus making it an Inner Product. The concept of an Inner Product Space is then introduced as a Real Vector Space equipped with an Inner Product. A second example involving an integral over the space of real polynomials is then explored. In the second part of the video Orthonormal Sets are introduced via a definition and then the proof of a lemma stating that any Orthonormal Set in an Inner Product Space is Linearly Independent. The video concludes with a final definition of a Sesquilinear Form and a discussion of a Complex Inner Product Space. 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://www.youtube.com/watch?v=9pF__coVyEE Calculating the inverse of 2x2, 3x3 and 4x4 matrices: https://www.youtube.com/watch?v=VKOaG3Ogf9Q What is the Determinant Function: https://www.youtube.com/watch?v=bLsBWVYSg0A The Easiest Method to Calculate Determinants: https://www.youtube.com/watch?v=qniUv4EZB0w Eigenvalues and Eigenvectors Explained: https://www.youtube.com/watch?v=8uISh6xyW7w Spectral Theorem Proof: https://www.youtube.com/watch?v=ADwsk9G5s_8 Vector Space Axioms: https://www.youtube.com/watch?v=draqOOUoWQM Subspace Test: https://www.youtube.com/watch?v=3_MxBlWQsgs Basis, Spanning and Linear Independence: https://www.youtube.com/watch?v=wdOFi8aUNp0 Dimension Formula: https://www.youtube.com/watch?v=lGo9E5dZ4f8 Direct Sum: https://www.youtube.com/watch?v=b1VZm4Oecq4 Linear Transformations: https://www.youtube.com/watch?v=UF59Mok4fsQ Rank Nullity Theorem: https://www.youtube.com/watch?v=yZ52iqxaiCc Produced by Dr Tom Crawford at the University of Oxford. Tom is Public Engagement Lead at the Oxford University Department of Continuing Education: https://www.conted.ox.ac.uk/ 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 Check out Proprep with a 30-day free trial here: https://www.proprep.uk/info/TOM-Crawford