Khan Academy: Linear Algebra: Projection Is Closest Vector in Subspace

Video gives examples of vector addition, graphing vectors, multiplying vectors by a scalar and vector subtraction using vectors in R2 and extends to vectors in other dimensions. [25:33]

A video lesson proving that the projection of a vector is actually the closest vector in the subspace to the original vector.

Demonstrates how to find the appropriate combination of two given vectors in order to obtain a third given vector. This is done by representing the problem with a single matrix equation and solving that equation. [14:20]

Video giving another reason why orthonormal bases are useful. First reviews how to find the projection of a vector onto a subspace and then describes how an orthonormal basis can be used to find the projection. Simplifies the...

Video defining a specific orthonormal basis and using the simplified formula found in the previous video to calculate the projection of a vector onto a subspace with the given orthonormal basis. [6:42]

A video lesson proving that any projection onto a subspace is actually a linear transformation. Also, if given the basis for the subspace then the specific transformation matrix for the projection can be found. Includes a brief...

A video lesson defining projections onto subspaces. Uses the definition for a projection onto a line and then shows that a generalized version of the formula works for a projection onto any subspace. Includes a concrete example and...

Video explains what a linear combination is and gives a concrete example of a linear combination of two vectors in two dimensions. Linear combinations are then extended to introduce spans of vectors. Shows a proof that two non-parallel...

Video starts with reviewing matrix notation. Defines multiplying a matrix by a vector and shows examples. Shows how the matrix-vector products can be interpreted as the dot product of column vectors with a matrix or as a weighted linear...

Video demonstrating that the area of a parallelogram created by graphing the column vectors of a 2x2 matrix is the equivalent to the absolute value of the determinant of the matrix. Pythagorean theorem, in conjunction with calculating a...