In this linear algebra lecture we introduce the concept of a vector space, consider the Cauchy-Schwartz inequality, triangle inequality, and do several problems with lines and planes.
In this lecture we learn about what a vector space over a field is, a subspace, a theorem on subspaces, inner product spaces, and do many examples on these topics.
In this lecture we study orthogonality, the inner product, dot product, inner product spaces, orthogonal matrices, QR decomposition, and the Gram-Schmidt algorithm.
In this lecture we learn about linear dependence, linear independence, basis of a vector space, dimension of a vector space, the row reduced echelon form of a matrix, the rank of a matrix, the null-space of a matrix, and the Wronskian.
In this lecture we study linear transformations, domain, codomain, image of a linear map f, kernel of a linear map f, vector space homomorphism, vector space isomorphism, fundamental subspaces, rank and nullity, rank(A)+nullity(A) = dimension of the domain of the linear map f. We give several examples taking the row-reduced echelon form of a matrix and getting a basis for the image of f and the kernel of f.
In this lecture we study eigenvalues and eigenvectors of matrices and use these to diagonalize a matrix and hence calculate high powers of the matrix.
In this lecture we cover exam practice exercises, examples, and revision of content introduced in Linear Algebra Lectures 1 through 6.
In this lecture we do a linear algebra exam in 42 minutes. This covers linear independence, basis, dimension, span, row space, column space, basis for the image of a linear map, basis for the kernel of a linear map, eigenvalue and eigenvectors, and diagonalization.
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