Video first graphs the transformation image of a specific transformation. Then examines the solution set for a couple particular points in the image. Shows that the solution set is a shifted version of the null space. [16:34]

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 shows that the transformation matrix of the composition of linear functions is the product of the transformation matrices of each linear transformation in the composition. [16:30]

Video continuing the example from the previous video where a transformation matrix with respect to an alternate basis was found given a transformation matrix with respect to the standard basis, an alternate basis and a change of basis...

Video first shows an intuitive sense of what composing linear transformations does and the notation used. Shows that the composition of linear transformations is also a linear transformation. [12:21]

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 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 shows how to simplify the vector projection formula using unit vectors. Shows that projections are linear transformations. Shows the general formula for the transformation matrix for projections in R^2. Shows an example using the...

Video shows how to simplify the vector projection formula using unit vectors. Shows that projections are linear transformations. Shows the general formula for the transformation matrix for projections in R^2. Shows an example using the...

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 reviewing the relationship between transformation matrices with respect to different bases. Addresses manipulating the equation to write it in terms of the transformation matrix with respect to the standard basis. The relationship...

Video first shows visually that a rotation is a linear transformation. Shows how to construct a transformation matrix that will rotate a vector through a given angle. Shows both the general rotation matrix and an example of a rotation...

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 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 first shows that matrix vector multiplication is another way to represent a transformation and then shows that matrix vector products are linear transformations. Relates vector transformations to video game graphics. This video...

Video first shows that matrix vector multiplication is another way to represent a transformation and then shows that matrix vector products are linear transformations. Relates vector transformations to video game graphics. This video...

Video first introduces the identity matrix and shows that when any vector is multiplied by the identity matrix the result is the original vector. Also shows that the columns of the identity matrix are called the standard basis for R^n....