Khan Academy: Linear Algebra: A More Formal Understanding of Functions

A more formal understanding of functions.

Video first reviews what it means for a function to be invertible. Shows that if a function is invertible then there is a unique solution to f(x) = y. Also shows that if there is a unique solution to the equation f(x) = y for all y then...

Video first reviews the meaning of functions and talks about functions mapping sets from R^n to R^m. Gives examples of writing functions in vector notation. Explains that in linear algebra a transformation is just a function operation on...

Video starts with reviewing matrix notation. Defines multiplying a matrix by a vector and shows examples. Shows how the matrix-vector products can be interpreted as the dot product of column vectors with a matrix or as a weighted linear...

Video first gives the conditions that define a linear transformation. Gives examples of determining if a transformation is linear or not. Shows several types of notation for vector transformations. [13:52]

Video explains what a surjective or onto function is and also what an injective or one-to-one function is. Gives examples and non-examples of surjective and injective functions. [9:31]

An instructional video that gives an example of how to find a solution to the equation Ax = b. In the example, the matrix A is 2 x 2. Shows that when the solution is a member of the row space of A for a given vector b, a member of the...

Video first reviews what a function is. Explains what the identity function is and uses it to define what makes a function invertible. Shows that the inverse will be unique. This video also appears in the strand Algebra: Functions. [18:53]

Video demonstrating that the area of a parallelogram created by graphing the column vectors of a 2x2 matrix is the equivalent to the absolute value of the determinant of the matrix. Pythagorean theorem, in conjunction with calculating a...

Video shows that the distributive property is true for multiplying matrices. [9:52]