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Inverse matrices, column space and null space: Essence of Linear Algebra - Part 7 of 15
How do you think about the column space and null space of a matrix visually? How do you think about the inverse of a matrix?
Khan Academy
Khan Academy: Linear Algebra: Null Space and Column Space Basis
Video first gives examples of finding the column space and null space of a matrix. Shows how to determine if the column space is a basis. Gives an example of finding the basis of the column space if the column space is not already a...
Khan Academy
Khan Academy: Linear Algebra: Null Space 2: Calculating the Null Space of Matrix
Video demonstrates how to calculate the null space of a matrix by creating a system of equations and using reduced row echelon form to solve and represent the null space by a linear combination of vectors and by the span of vectors. This...
Khan Academy
Khan Academy: Linear Algebra: Exploring the Solution Set of Ax=b
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]
Khan Academy
Khan Academy: Linear Algebra: Dimension of the Null Space or Nullity
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...
Khan Academy
Khan Academy: Linear Algebra: Null Space 3: Relation to Linear Independence
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...
Khan Academy
Khan Academy: Linear Algebra: Unique Row Space Solution to Ax = B
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...
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Khan Academy: Linear Algebra: Visualizations of Left Nullspace and Rowspace
Video builds upon the column spaces and null spaces of the 2x3 matrix and its transpose found in the previous video. Uses them to explain some relationships between the four fundamental subspaces, namely that the orthogonal complement of...
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Khan Academy: Linear Algebra: Introduction to the Null Space of a Matrix
Video first reviews the criteria for a subspace and then goes through the steps to show the null space meets the criteria and is a a subspace. This video also appears in the strand Algebra: Matrices. [10:21]
Khan Academy
Khan Academy: Linear Algebra: Matrix Condition for One to One Trans
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...
Khan Academy
Khan Academy: Linear Algebra: Rowspace and Left Nullspace
Video reviewing two fundamental subspaces of a matrix (null space and column space) and introducing two more (row space and left null space). Finds each of these subspaces for a specific 2x3 matrix and its transpose. Demonstrates that...
Khan Academy
Khan Academy: Linear Algebra: Row Space Solution to Ax = B Example
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...
Khan Academy
Khan Academy: Linear Algebra: Orthogonal Complements as Subspaces
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...
Khan Academy
Khan Academy: Linear Algebra: Visualizing a Column Space as a Plane in R3
Video reviews how to find a basis for the column space and talks about how to find it more efficiently using reduced row echelon form. Shows a visualization of the column space in R3 as a plane. Shows two methods to determine the...
Khan Academy
Khan Academy: Linear Algebra: Null Space 3: Relation to Linear Independence
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...
Khan Academy
Khan Academy: Orthogonal Complements: Unique Rowspace Solution to Ax=b
This video shows that, for any b that is in the column space of A, there is a unique member of the row space that is the "smallest" solution to Ax=b.
Khan Academy
Khan Academy: Orthogonal Complements: Orthogonal Complement of the Nullspace
A video lesson illustrating that the orthogonal complement of the nullspace is equal to the row space and vice versa.
Khan Academy
Khan Academy: Transpose of a Matrix: Rowspace and Left Nullspace
Students explore transformations of matrices, rowspace and nullspace by watching a video tutorial.
Khan Academy
Khan Academy: Linear Algebra: Row Space Solution to Ax = B Example
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...
Khan Academy
Khan Academy: Orthogonal Complements: Orthogonal Complements
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...
Khan Academy
Khan Academy: Linear Algebra: Visualizing a Column Space as a Plane in R3
Video reviews how to find a basis for the column space and talks about how to find it more efficiently using reduced row echelon form. Shows a visualization of the column space in R3 as a plane. Shows two methods to determine the...
Khan Academy
Khan Academy: Linear Algebra: Visualizing a Column Space as a Plane in R3
Video reviews how to find a basis for the column space and talks about how to find it more efficiently using reduced row echelon form. Shows a visualization of the column space in R3 as a plane. Shows two methods to determine the...
Khan Academy
Khan Academy: Defining a Plane in R3 With a Point and Normal Vector
This video shows the equation for a plane in R3 using a point on the plane and a normal vector.