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
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
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
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
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
About this unit
Understanding how we can map one set of vectors to another set. Matrices used to define linear transformations.