Khan Academy: Linear Algebra: Image of a Subset Under a Transformation

Video shows an example of the image of a subset of a domain under a linear transformation. Example includes defining a triangle using vectors. [18:11]

Video demonstrating that the area of a parallelogram that is the image of a rectangle under a transformation is equivalent to the absolute value of the determinant of the matrix whose column vectors generate the parallelogram. Includes...

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 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 showing how to change coordinate systems to make finding the transformation matrix with respect to the standard basis easier. First, a specific linear transformation is defined and the regular method for constructing the...

Video first reviews what the image under a transformation is and then introduces the concept of preimage. [5:23]

Video first shows how to find the preimage of a set under a transformation. Shows visually what the preimage looks like on a graph. Explains that the kernel is the preimage of the zero vector. [15:23]

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...

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 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...