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Functions and linear transformations

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A more formal understanding of functions

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Vector transformations

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Linear transformations

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Matrix vector products as linear transformations

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Linear transformations as matrix vector products

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Image of a subset under a transformation

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im(T): Image of a transformation

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Preimage of a set

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Preimage and kernel example

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Sums and scalar multiples of linear transformations

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More on matrix addition and scalar multiplication

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Linear transformation examples

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Linear transformation examples: Scaling and reflections

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Linear transformation examples: Rotations in R2

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Rotation in R3 around the x-axis

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Unit vectors

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Introduction to projections

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Expressing a projection on to a line as a matrix vector prod

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Transformations and matrix multiplication

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Compositions of linear transformations 1

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Compositions of linear transformations 2

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Matrix product examples

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Matrix product associativity

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Distributive property of matrix products

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Inverse functions and transformations

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Introduction to the inverse of a function

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Proof: Invertibility implies a unique solution to f(x)=y

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Surjective (onto) and injective (one-to-one) functions

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Relating invertibility to being onto and one-to-one

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Determining whether a transformation is onto

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Exploring the solution set of Ax = b

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Matrix condition for one-to-one transformation

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Simplifying conditions for invertibility

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Showing that inverses are linear

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Finding inverses and determinants

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Deriving a method for determining inverses

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Example of finding matrix inverse

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Formula for 2×2 inverse

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3 x 3 determinant

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n x n determinant

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Determinants along other rows/cols

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Rule of Sarrus of determinants

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More determinant depth

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Determinant when row multiplied by scalar

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(correction) scalar multiplication of row

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Determinant when row is added

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Duplicate row determinant

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Determinant after row operations

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Upper triangular determinant

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Simpler 4×4 determinant

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Determinant and area of a parallelogram

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Determinant as scaling factor

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Transpose of a matrix

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Transpose of a matrix

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Determinant of transpose

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Transpose of a matrix product

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Transposes of sums and inverses

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Transpose of a vector

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Rowspace and left nullspace

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Visualizations of left nullspace and rowspace

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rank(a) = rank(transpose of a)

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Showing that A-transpose x A is invertible

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Orthogonal complements

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Orthogonal complements

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dim(v) + dim(orthogonal complement of v) = n

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Representing vectors in rn using subspace members

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Orthogonal complement of the orthogonal complement

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Orthogonal complement of the nullspace

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Unique rowspace solution to Ax = b

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Rowspace solution to Ax = b example