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Linear algebra for advanced studies
·
Module 2
Linear Transformations and Matrices
1
Introduction to Linear Transformations
Define linear transformations and determine if a given function is linear.
2
Matrix Representation of Linear Transformations
Find the matrix representation of a linear transformation with respect to given bases.
3
Null Space and Column Space of a Matrix
Compute the null space (kernel) and column space (image) of a matrix.
4
Rank-Nullity Theorem: Dimensions of Subspaces
Apply the rank-nullity theorem to relate the dimensions of the domain, kernel, and image.
5
Invertibility of Linear Transformations from Matrices
Determine the invertibility of a linear transformation from its matrix representation.
6
Change-of-Basis Matrices
Construct the change-of-basis matrix between two bases.
7
Matrix Transformations Under Basis Change
Compute how a matrix representation changes under a change of basis.
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Foundations of Vector Spaces
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Inner Products and Orthogonality