The Four Fundamental Subspaces and the Fundamental Theorem | Linear Algebra
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We introduce the four fundamental
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
Column Spaces of a Matrix Explained: Column Space of a Matrix Explained | ...
Find Basis for the Row Space: Finding Basis for the Row Space of a ...
Find Basis for the Column Space: Finding Basis for the Column Space of...
Find Null Space and Nullity of a Matrix: Find Null Space and Nullity of a Matr...
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Linear Algebra course: Linear Algebra
Linear Algebra exercises: Linear Algebra Exercises
Get the textbook for this course! https://amzn.to/43xAWEz
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
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Column Spaces of a Matrix Explained: Column Space of a Matrix Explained | ...
Find Basis for the Row Space: Finding Basis for the Row Space of a ...
Find Basis for the Column Space: Finding Basis for the Column Space of...
Find Null Space and Nullity of a Matrix: Find Null Space and Nullity of a Matr...
Join Wrath of Math to get exclusive videos, lecture notes, and more:
@wrathofmath
Linear Algebra course: Linear Algebra
Linear Algebra exercises: Linear Algebra Exercises
Get the textbook for this course! https://amzn.to/43xAWEz
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
★DONATE★
◆ Support Wrath of Math on Patreon: Patreon: wrathofmathlessons
◆ Donate on PayPal: https://www.paypal.me/wrathofmath
Outro music by Ben Watts and is available for channel members.
Follow Wrath of Math on...
● Instagram: Instagram: wrathofmathedu
● TikTok: TikTok: wrathofmathedu
● X: https://x.com/wrathofmathedu
● Facebook: Facebook: WrathofMath
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