Curated Video
Data Science and Machine Learning (Theory and Projects) A to Z - Mathematical Foundation: SubSpace
In this video, we will cover SubSpace. This clip is from the chapter "Machine Learning: Feature Engineering and Dimensionality Reduction with Python" of the series "Data Science and Machine Learning (Theory and Projects) A to Z".In this...
Khan Academy
Khan Academy: Linear Algebra: Projections Onto Subspaces
A video lesson defining projections onto subspaces. Uses the definition for a projection onto a line and then shows that a generalized version of the formula works for a projection onto any subspace. Includes a concrete example and...
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Khan Academy: Linear Algebra: Projection Is Closest Vector in Subspace
A video lesson proving that the projection of a vector is actually the closest vector in the subspace to the original vector. [9:05]
Khan Academy
Khan Academy: Linear Algebra: Projection Onto a Subspace Is a Lin Transformation
A video lesson proving that any projection onto a subspace is actually a linear transformation. Also, if given the basis for the subspace then the specific transformation matrix for the projection can be found. Includes a brief...
Khan Academy
Khan Academy: Linear Algebra: Introduction to the Null Space of a Matrix
Video first reviews the criteria for a subspace and then goes through the steps to show the null space meets the criteria and is a a subspace. This video also appears in the strand Algebra: Matrices. [10:21]
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Khan Academy: Linear Algebra: Representing Vectors in Rn Using Subspace Members
A video lesson showing a series of proofs that culminate in explaining that any member of Rn can be represented as a unique sum of a member of some subspace of Rn and a member of its orthogonal complement. Ideas included in the proof are...
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Khan Academy: Linear Algebra: Linear Subspaces
Video defines a vector subspace by its three criteria: containing the zero vector, closure under scalar multiplication and closure under vector addition. Gives examples of vector sets and testing to see if they are vector subspaces. Then...
Khan Academy
Khan Academy: Linear Algebra: Subpsace Projection Matrix Example
An instructional video that works through a concrete example for how to find the projection of an arbitrary vector onto a specific subspace in R4. Uses a 4 x 2 basis matrix for the subspace. Matrix operations are used along with the...
Khan Academy
Khan Academy: Linear Algebra: Visualizing a Projection Onto a Plane
A video lesson showing what a projection onto a plane could look like. Illustrates that the newly derived definition of a projection holds true for projections onto subspaces other than lines. [9:27]
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Khan Academy: Linear Algebra: Another Example of a Projection Matrix
A video lesson working another example for finding the projection of an arbitrary vector onto a specific subspace in R3. The basis of the subspace is a 3 x 2 matrix. Uses an alternate method for find the transformation matrix that...
Khan Academy
Khan Academy: Linear Algebra: Orthogonal Complement of the Orthogonal Complement
A video proving that the orthogonal complement of the orthogonal complement of some subspace V must be a member of the original subspace V. Proves that V and the orthogonal complement of the orthogonal complement of V are exactly the...
Khan Academy
Khan Academy: Orthogonal Projections: Projection Is Closest Vector in Subspace
A video lesson proving that the projection of a vector is actually the closest vector in the subspace to the original vector.
Khan Academy
Khan Academy: Linear Algebra: Im(t): Image of a Transformation
Video first reviews the closure properties of subspaces. Shows that the image of a subspace under a linear transformation is a subspace. Connects the meaning of range to the image of the transformation. Shows that the image of the linear...
Khan Academy
Khan Academy: Linear Algebra: Im(t): Image of a Transformation
Video first reviews the closure properties of subspaces. Shows that the image of a subspace under a linear transformation is a subspace. Connects the meaning of range to the image of the transformation. Shows that the image of the linear...
Khan Academy
Khan Academy: Linear Algebra: Representing Vectors in Rn Using Subspace Members
A video lesson showing a series of proofs that culminate in explaining that any member of Rn can be represented as a unique sum of a member of some subspace of Rn and a member of its orthogonal complement. Ideas included in the proof are...
Khan Academy
Khan Academy: Representing Vectors in Rn Using Subspace Members
This video shows that any member of Rn can be represented as a unique sum of a vector in subspace V and a vector in the orthogonal complement of V.
Khan Academy
Khan Academy: Linear Algebra: Linear Subspaces
Video defines a vector subspace by its three criteria: containing the zero vector, closure under scalar multiplication and closure under vector addition. Gives examples of vector sets and testing to see if they are vector subspaces. Then...
Khan Academy
Khan Academy: Subspaces and the Basis for a Subspace: Linear Subspaces
Introduction to linear subspaces. This video gives examples of vector sets and testing to see if they are vector subspaces.
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Khan Academy: Orthogonal Projections: Subspace Projection Matrix Example
This video uses a concrete example for how to find the projection of an arbitrary vector onto a specific subspace in R4. Uses a 4 x 2 basis matrix for the subspace.
Khan Academy
Khan Academy: Orthogonal Projections: Another Example of a Projection Matrix
A video lesson figuring out the transformation matrix for a projection onto a subspace by figuring out the matrix for the projection onto the subspace's orthogonal complement first.
Khan Academy
Khan Academy: Linear Algebra: Orthogonal Complement of the Orthogonal Complement
A video proving that the orthogonal complement of the orthogonal complement of some subspace V must be a member of the original subspace V. Proves that V and the orthogonal complement of the orthogonal complement of V are exactly the...
Khan Academy
Khan Academy: Orthogonal Complement of the Orthogonal Complement
A video proving that the orthogonal complement of the orthogonal complement of some subspace V must be a member of the original subspace V.
Khan Academy
Khan Academy: Orthogonal Complements: Dim(v) + Dim(orthogonal Complement of V)=n
Showing that if V is a subspace of Rn, then dim(V) + dim(V's orthogonal complement) = n