The Fundamental Theorem of Linear Algebra

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In this video, we present the Fundamental Theorem of Linear Algebra, which provides a connection between the four fundamental spaces of a matrix, as well as their dimensions as in the usual Rank-Nullity Theorem, and we see them as orthogonal complements and orthogonal duals. We also demonstrate how to find the general solution of a non-homogenous linear system where the particular solution is contained in the row space and hence is the shortest solution to the linear system.

Linear Algebra Done Openly is an open source linear algebra textbook developed by Dr. Andrew Misseldine. It is supported and improved by student contributions done by Math 2700 students at Southern Utah University, since Fall 2018. The textbook can be found at:

This lecture covers section 4.6 - The Fundamental Theorem of Linear Algebra from the book. This is part 3 of 3 videos from this section.

Andrew Misseldine

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We used the dot product with the base for the null space rather than differentiation to optimise x-row. Refreshing, and I am sure this idea will be carried forwards, dot instead of differentiate. Another perspective on calculus optimisation. Thank you

darrenpeck

Hi, that is a very nice explanation. Why the negation is 1 in our example?

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