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://youtu.be/UF59Mok4fsQ
Spanning, Basis & 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-%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://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
Linear Transformations: https://youtu.be/UF59Mok4fsQ
Rank Nullity Theorem: https://youtu.be/yZ52iqxaiCc
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