Video shows an example of proving that a particular linear combination of 3 vectors in R3 spans R3. Relates this to linearly independent and linearly dependent vectors. [16:53]

Video expands the examples of linearly independent and linearly dependent vectors shown in the previous video to give a formal definition of linearly dependent vectors. Shows a proof of the definition and then gives an example of using...

Video reviews what the null space of a matrix is and how to calculate it, then relates it to linear independence. Shows that if the vectors are linearly independent then the null space is the zero vector. Also shows that if the null...

Video explains what linearly dependent vectors and linearly independent vectors are. Shows visual representations to help explain the concept and relates it to the span of the linear combination of the vectors. [15:46]

Video first reviews what the null space is and how to solve for it using reduced row echelon form. Shows that any solution to the inhomogeneous system Ax = b will be the sum of a particular solution and a homogeneous solution. Extends...

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...

A video lesson showing why the rank of a matrix is the equal to the rank of that matrix's transpose. [11:13]

Video explains what a basis for a vector subspace is and its criteria: the vectors must span the subspace and be linearly independent. Also gives an informal definition of basis as the minimum set of vectors that spans the subspace....

Video shows an example of proving a particular linear combination of 3 vectors in R3 spans R3. Relates this to linearly independent and linearly dependent vectors. [16:53]

Video expands the examples of linearly independent and linearly dependent vectors shown in the previous video to give a formal definition of linearly dependent vectors. Shows a proof of the definition and then gives an example of using...

Video reviews what the null space of a matrix is and how to calculate it, then relates it to linear independence. Shows that if the vectors are linearly independent then the null space is the zero vector. Also shows that if the null...

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...

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.

A video lesson showing why the rank of a matrix is the equal to the rank of that matrix's transpose. [11:13]