The Four Fundamental Subspaces and the Fundamental Theorem | Linear Algebra

Support the production of this course by joining Wrath of Math to access all my Linear Algebra videos plus lecture notes at the premium tier!    / @wrathofmath   🛍 Check out my math fashion brand! https://mathshion.com/ Linear Algebra course:    • Linear Algebra   Linear Algebra exercises:    • Linear Algebra Exercises   Get the textbook for this course! https://amzn.to/45KYgmA Business Inquiries: [email protected] We introduce the four fundamental spaces associated with an mxn matrix A. These are the row space of A, the column space of A, the null space of A, and the null space of A transpose (also called the left null space of A). The row space and the null space of A are subspaces of R^n. The column space of A and the null space of A^T are subspaces of R^m. We'll then investigate the relationship between these spaces, and see how their dimensions can all be determined from the size of the matrix A and the rank of A. We then see that the fundamental spaces of the matrix come in orthogonal pairs, and in total we prove the fundamental theorem of linear algebra. #linearalgebra ★DONATE★ ◆ Support Wrath of Math on Patreon:   / wrathofmathlessons   ◆ Donate on PayPal: https://www.paypal.me/wrathofmath Follow Wrath of Math on... ● Instagram:   / wrathofmathedu   ● TikTok:   / wrathofmathedu   ● X: https://x.com/wrathofmathedu 0:00 Intro 0:34 Row Space, Column Space, and Null Space 1:38 The Four Fundamental Spaces 3:59 Subspaces of R^? 6:11 The Dimensions of the Subspaces 10:51 Spaces as Orthogonal Complements 19:25 The Fundamental Theorem of Linear Algebra 21:02 Conclusion