Pages similar to: Matrices and linear transformations
- Linear transformations
Definition of a linear transformation of Euclidean spaces. - The derivative matrix
The derivative matrix is presented as a natural generalization of the single variable derivative to multivariable functions. - Determinants and linear transformations
A description of how a determinant describes the geometric properties of a linear transformation. - How linear transformations map parallelograms and parallelepipeds
Why linear transformations map parallelograms onto parallelograms and parallelepipeds onto parallelepipeds. - Matrices and determinants for multivariable calculus
Brief description of matrices and determinants, at the level needed for multivariable calculus. - Multiplying matrices and vectors
How to multiply matrices with vectors and other matrices. - Matrix and vector multiplication examples
Examples demonstrating how to multiply matrices and vectors. - Introduction to matrices
A brief introduction to matrices. - The transpose of a matrix
Definition of the transpose of a matrix or a vector. - Dot product in matrix notation
How to view the dot product between two vectors as a product of matrices. - The definition of differentiability in higher dimensions
The definition of differentiability for multivariable functions. Informal derivation designed to give intuition behind the condition for a function to be differentiable. - Subtleties of differentiability in higher dimensions
A description of some of the tricky ways where a function of multiple variables can fail to be differentiable. Example two variable functions are illustrated with interactive graphics. - Area calculation for changing variables in double integrals
A derivation of how a mapping that changes variables in double integrals transforms area. - Geometric properties of the determinant
Explanation of some of the basic geometric properties of the determinant. - Polar coordinates mapping
How polar coordinates can be viewed as mapping from the polar plane onto the Cartesian plane. - Volume calculation for changing variables in triple integrals
A derivation of how a mapping that changes variables in triple integrals transforms volume. - The linear function
Overview of the linear function of one variable and a few of its properties.
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