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# Matrix transformations

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A more formal understanding of functionsVector transformationsLinear transformationsVisualizing linear transformationsMatrix from visual representation of transformationMatrix vector products as linear transformationsLinear transformations as matrix vector productsImage of a subset under a transformationim(T): Image of a transformationPreimage of a setPreimage and kernel exampleSums and scalar multiples of linear transformationsMore on matrix addition and scalar multiplication

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Linear transformation examples: Scaling and reflectionsLinear transformation examples: Rotations in R2Rotation in R3 around the x-axisUnit vectorsIntroduction to projectionsExpressing a projection on to a line as a matrix vector prod

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Compositions of linear transformations 1Compositions of linear transformations 2Matrix product examplesMatrix product associativityDistributive property of matrix products

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Introduction to the inverse of a functionProof: Invertibility implies a unique solution to f(x)=ySurjective (onto) and injective (one-to-one) functionsRelating invertibility to being onto and one-to-oneDetermining whether a transformation is ontoExploring the solution set of Ax = bMatrix condition for one-to-one transformationSimplifying conditions for invertibilityShowing that inverses are linear

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Deriving a method for determining inversesExample of finding matrix inverseFormula for 2x2 inverse3 x 3 determinantn x n determinantDeterminants along other rows/colsRule of Sarrus of determinants

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Determinant when row multiplied by scalar(correction) scalar multiplication of rowDeterminant when row is addedDuplicate row determinantDeterminant after row operationsUpper triangular determinantSimpler 4x4 determinantDeterminant and area of a parallelogramDeterminant as scaling factor

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Transpose of a matrixDeterminant of transposeTranspose of a matrix productTransposes of sums and inversesTranspose of a vectorRowspace and left nullspaceVisualizations of left nullspace and rowspacerank(a) = rank(transpose of a)Showing that A-transpose x A is invertible