Column Matrix Teacher Resources

Video introducing the orthogonal complement of a subspace and showing that it is also a subspace. Proves that the null space of a matrix is equivalent to the orthogonal complement of its row space by showing that every member of the null...

Video reviews how to find the null space of a matrix using reduced row echelon form. Shows that the vectors of the null space are linearly independent and form a basis for the null space. Extends this to show how the nullspace can be...

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...

A video lesson proving that there is a unique solution in the row space of A to the equation Ax = b, where A is some m x n matrix, b is a vector that is a particular member of the column space of A, and x is a member of Rn. The solution...

Understanding how the null space of a matrix relates to the linear independence of its column vectors. [11:34]

A "minor" is the determinant of the square matrix formed by deleting one row and one column from some larger square matrix. To find the determinant of the matrix A, you have to pick a row or a column of the matrix, find all the cofactors...

Video reviews what the null space of a matrix is and how to calculate it, then relates it to linear independence. Shows that if the vectors are linearly independent then the null space is the zero vector. Also shows that if the null...

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...

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 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.

Video introducing the orthogonal complement of a subspace and showing that it is also a subspace. Proves that the null space of a matrix is equivalent to the orthogonal complement of its row space by showing that every member of the null...

Dozens of statistical terms are defined and illustrated in this glossary.

Video shows proof that any basis of a particular subspace has the same number of elements. Leads up to give the definition of dimension.

This video shows that linear independence of pivot columns implies linear independence of the corresponding columns in the original equation.

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 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]

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 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...

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...

This video shows the equation for a plane in R3 using a point on the plane and a normal vector.

This is a worksheet in a pdf format on Semantic Feature Analysis. The worksheet, or matrix, is divided into boxes in three rows with four columns across.