To add the vectors (x₁,y₁) and (x₂,y₂), we add the corresponding components from each vector: (x₁+x₂,y₁+y₂). Here's a concrete example: the sum of (2,4) and (1,5) is (2+1,4+5), which is (3,9). There's also a nice graphical way to add vectors, and the two ways will always result in the same vector.​​

Sal discusses the conditions of matrix dimensions for which addition or multiplication are defined. Created by Sal Khan.

Overview of the Intermediate Value Theorem, the Extreme Value Theorem and the Mean Value Theorem.

We've learned about matrix addition, matrix subtraction, matrix multiplication. So you might be wondering, is there the equivalent of matrix division? And before we get into that, let me introduce some concepts to you. And then we'll see that there is something that maybe isn't exactly division, but it's analogous to it.

Sal checks whether the commutative property applies for matrix multiplication. In other words, he checks whether for any two matrices A and B, A*B=B*A (the answer is NO, by the way). Created by Sal Khan.

This 10-minute video lesson looks at Determinants: Finding the determinant of a 3x3 matrix.

This 22-minute video lesson shows another example of a projection matrix. It shows how to figure out the transformation matrix for a projection onto a subspace by figuring out the matrix for the projection onto the subspace's orthogonal complement first.

This 15-minute video lesson sees that orthonormal bases make for good coordinate systems.

This 3-minute video lesson is a correction of last video showing that the determinant when one row is multiplied by a scalar is equal to the scalar times the determinant.

This 14-minute video lesson shows how to determine the equation for a plane in R3 using a point on the plane and a normal vector.

This 25-minute video lesson introduces the idea of an angle between two vectors.

This 10-minute video lesson looks at what happens to the determinant when we perform a row operation.

This 13-minute video lesson looks at the determinant when a row is multiplied by a scalar.

This 17-minute video lesson looks at the determinant when one matrix has a row that is the sum of the rows of other matrices (and every other term is identical in the 3 matrices).

This 22-minute video lesson realizes that the determinant of a 2x2 matrix is equal to the area of the parallelogram defined by the column vectors of the matrix.

This 20-minute video lesson shows how to view the determinant of the transformation matrix as a scaling factor of regions.

This 14-minute video lesson shows the proof by induction that transposing a matrix does not change its determinant.