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:    • Oxford Linear Algebra: Linear Transfo...  
Spanning, Basis & Linear Independence:    • Oxford Linear Algebra: Basis, Spannin...  

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/downlo...

You can also find several video lectures from ProPrep explaining the content covered in the video here: https://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 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):    • Oxford Linear Algebra: Elementary Row...  

Calculating the inverse of 2x2, 3x3 and 4x4 matrices:    • Oxford Linear Algebra: How to find a ...  

What is the Determinant Function:    • Oxford Linear Algebra: What is the De...  

The Easiest Method to Calculate Determinants:    • Oxford Linear Algebra: The Easiest Me...  

Eigenvalues and Eigenvectors Explained:    • Oxford Linear Algebra: Eigenvalues an...  

Spectral Theorem Proof:    • Oxford Linear Algebra: Spectral Theor...  

Vector Space Axioms:    • Oxford Linear Algebra: What is a Vect...  

Subspace Test:    • Oxford Linear Algebra: Subspace Test  

Basis, Spanning and Linear Independence:    • Oxford Linear Algebra: Basis, Spannin...  

Dimension Formula:    • Oxford Linear Algebra: Dimension Form...  

Direct Sum:    • Oxford Linear Algebra: Direct Sum of ...  

Linear Transformations:    • Oxford Linear Algebra: Linear Transfo...  

Rank Nullity Theorem:    • Oxford Linear Algebra: Rank Nullity T...  

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/

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Check out Proprep with a 30-day free trial here: https://www.proprep.uk/info/TOM-Crawford