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Orthogonal and Orthonormal Bases | Linear Algebra
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We introduce orthogonal bases and orthonormal bases for inner product spaces. We'll prove an important theorem stating orthogonal sets of vectors are linearly independent, and see several examples of orthogonal and orthonormal bases. We'll also prove a theorem telling us how to write a vector as a linear combination of orthogonal basis vectors. #linearalgebra
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0:00 Intro
0:25 Definition of Orthogonal and Orthonormal Sets
0:56 Linear Independence of Orthogonal Sets
4:58 Definition of Orthogonal and Orthonormal Bases
5:32 Orthonormal Basis for R^n
6:03 Orthonormal Basis for Pn
7:50 Non-standard Orthonormal Basis for R^3
8:37 Writing a Vector as a Linear Combination of Orthonormal Basis Vectors
10:28 Proof for Orthonormal Basis Expression
13:26 Coordinate Vector Relative to Orthonormal Basis
16:20 Conclusion
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Thanks to Loke Tan, Raül Beienheimer, Matt Venia, Micheline, Doug Walker, Odd Hultberg, Marc, Shlome Ashkenazi, Barbora Sharrock, Mohamad Nossier, Rolf Waefler, Shadow Master, and James Mead for their generous support on Patreon!
Outro music is mine. You cannot find it anywhere, for now.
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Join Wrath of Math to get exclusive videos, music, and more:
0:00 Intro
0:25 Definition of Orthogonal and Orthonormal Sets
0:56 Linear Independence of Orthogonal Sets
4:58 Definition of Orthogonal and Orthonormal Bases
5:32 Orthonormal Basis for R^n
6:03 Orthonormal Basis for Pn
7:50 Non-standard Orthonormal Basis for R^3
8:37 Writing a Vector as a Linear Combination of Orthonormal Basis Vectors
10:28 Proof for Orthonormal Basis Expression
13:26 Coordinate Vector Relative to Orthonormal Basis
16:20 Conclusion
◉Textbooks I Like◉
★DONATE★
Thanks to Loke Tan, Raül Beienheimer, Matt Venia, Micheline, Doug Walker, Odd Hultberg, Marc, Shlome Ashkenazi, Barbora Sharrock, Mohamad Nossier, Rolf Waefler, Shadow Master, and James Mead for their generous support on Patreon!
Outro music is mine. You cannot find it anywhere, for now.
Follow Wrath of Math on...
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