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 reviews from previous videos that a function is invertible if and only if it is onto and one-to-one. Shows how to determine if a linear transformation is onto by showing that the columns of the transformation matrix span the...

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 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 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 first reviews what the null space is and how to solve for it using reduced row echelon form. Shows that any solution to the inhomogeneous system Ax = b will be the sum of a particular solution and a homogeneous solution. Extends...

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....

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....

Video first defines the sum of transformations. Shows that the sum of two transformations is equal to the sum of their transformation matrices. Defines scalar multiples of linear transformations and shows that they are equal to the...

Video shows example of multiplying matrices. Includes explanation of the dimensions needed in order for matrices to be able to be multiplied. Connects to the composition of linear transformations. This video also appears in the strand...

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

Video shows how to write a transformation matrix in R^3 that rotates a vector around the x-axis. Compares the rotation matrix in R^3 to the rotation matrix in R^2 found in the previous video and discusses how it could be applied for more...

Video first reviews how to find the transformation matrix using the identity matrix. Shows how to write a transformation matrix that reflects over the y-axis and scales the y-coordinates by a factor of 2. [15:13]

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]

Shows how to determine if a linear transformation is onto by showing that the columns of the transformation matrix span the co-domain.

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 first reviews the closure properties of subspaces. Shows that the image of a subspace under a linear transformation is a subspace. Connects the meaning of range to the image of the transformation. Shows that the image of the linear...