Khan Academy: Linear Algebra: Introduction to Linear Independence

Video defining an orthonormal set as a set of vectors that all have the length of 1 and are orthogonal to each other. Using proof by contradiction, the set is proven to be linearly independent and can be considered an orthonormal basis....

Video first reviews what a function is. Explains what the identity function is and uses it to define what makes a function invertible. Shows that the inverse will be unique. This video also appears in the strand Algebra: Functions. [18:53]

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]

Video reviews how to use vectors to define a line. Gives an intuitive sense of what a projection is, then explains step-by-step where the formula for projections comes from. Shows an example of calculating a vector projection. [14:37]

Video defining eigenvectors as scaled vectors and eigenvalues as their scale value and explaining their significance. Video references the example from a previous video about changing coordinate systems to make finding the transformation...

Video shows an example of proving that a particular linear combination of 3 vectors in R3 spans R3. Relates this to linearly independent and linearly dependent vectors. [16:53]

Video expands the examples of linearly independent and linearly dependent vectors shown in the previous video to give a formal definition of linearly dependent vectors. Shows a proof of the definition and then gives an example of using...

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 reviews how to use vectors to define a line. Gives an intuitive sense of what a projection is then explains step by step where the formula for projections comes from. Shows example of calculating vector projections. [14:37]

That old math tutor from is back and he is here to introduce the concept of algebra. This clip is actually quite clear and informative. Variables, constants, and algebra are defined along side solid examples. This clip would be good for...