Vectors are easier to understand when they're described in terms of orthogonal bases. In addition, the Four Fundamental Subspaces are orthogonal to each other in pairs.
If A is a rectangular matrix, Ax = b is often unsolvable. The matrix ATA will help us find a vector x̂ that comes as close as possible to solving Ax = b.
Orthogonal Vectors and Subspaces
Problem Solving: Orthogonal Vectors and Subspaces
Work the problems on your own and check your answers when you're done.