Khan Academy
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]
Khan Academy
Khan Academy: Linear Algebra: Showing That Inverses Are Linear
A video lesson that gives an in-depth, algebraic proof showing that the inverse of a linear transformation is also linear. [21:25]
Khan Academy
Khan Academy: Linear Algebra: Matrix Vector Products as Linear Transformations
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...
Khan Academy
Khan Academy: Linear Algebra: Matrix Vector Products as Linear Transformations
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...
Khan Academy
Khan Academy: Linear Algebra: Linear Transformations as Matrix Vector Products
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....
Khan Academy
Khan Academy: Linear Algebra: Linear Transformations as Matrix Vector Products
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....
Khan Academy
Khan Academy: Linear Algebra: Subpsace Projection Matrix Example
An instructional video that works through a concrete example for how to find the projection of an arbitrary vector onto a specific subspace in R4. Uses a 4 x 2 basis matrix for the subspace. Matrix operations are used along with the...
Khan Academy
Khan Academy: Linear Algebra:sums and Scalar Multiples of Linear Transformations
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...
Khan Academy
Khan Academy: Linear Algebra: Matrix Product Examples
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...
Khan Academy
Khan Academy: Linear Algebra: Another Example of a Projection Matrix
A video lesson working another example for finding the projection of an arbitrary vector onto a specific subspace in R3. The basis of the subspace is a 3 x 2 matrix. Uses an alternate method for find the transformation matrix that...
Khan Academy
Khan Academy: Linear Algebra: Simplifying Conditions for Invertibility
A video lesson explaining that a transformation matrix is invertible if and only if it is a square identity matrix in reduced row echelon form. This video is also found in the strand Algebra: Matrices. [6:37]
Khan Academy
Khan Academy: Linear Algebra: Linear Transformations
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]
Khan Academy
Khan Academy: Linear Algebra: Rotation in R3 Around the X Axis
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...
Khan Academy
Khan Academy: Linear Algebra: Linear Transformations: Scaling and Reflections
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]
Khan Academy
Khan Academy: Linear Algebra: Compositions of Linear Transformations 2
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]
Khan Academy
Khan Academy: A Projection Onto a Subspace Is a Linear Transforma
A video lesson proving that any projection onto a subspace is actually a linear transformation. It includes a brief description of how the results can be useful in 3-D graphical programming.
Khan Academy
Khan Academy: Determining Whether a Transformation Is Onto
Shows how to determine if a linear transformation is onto by showing that the columns of the transformation matrix span the co-domain.
Khan Academy
Khan Academy: Linear Algebra: Compositions of Linear Transformations 1
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]
Khan Academy
Khan Academy: More Determinant Depth: Determinant as Scaling Factor
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.
Khan Academy
Khan Academy: Linear Algebra: Im(t): Image of a Transformation
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...
Khan Academy
Khan Academy: Linear Algebra: Im(t): Image of a Transformation
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...
Khan Academy
Khan Academy: Image of a Subset Under a Transformation
Exploring what happens to a subset of the domain under a transformation
Khan Academy
Khan Academy: Orthogonal Projections: Subspace Projection Matrix Example
This video uses a concrete example for how to find the projection of an arbitrary vector onto a specific subspace in R4. Uses a 4 x 2 basis matrix for the subspace.
Khan Academy
Khan Academy: Linear Algebra:sums and Scalar Multiples of Linear Transformations
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...