vectors

How could you use the properties of the dot product to prove the following identities: (where u and v denote vectors in Rn) a) ||u + v||^2 + ||u-v||^2 = 2(||u||^2 + ||v||^2) b) ||u + v||^2 - ||u-v||^2 = 4u dot v Note: dot = dot product ^ = power ||= distance

vectors

Could you please give me some sort of a sketch or drawing of what a set S (its interior and its closure) would look like when: S= {(x,y) : -1<= y < cosx, -2# < x <= 2#} Note: <= is less than or equal to < is less than # is pi (3.14....)

vectors that are linearly independent

If the set{v1,v2,v3} of vectors in R^m is lineraly dependent, how do I know that the set {v1,v2,v3,v4} is also lineraly dependent for every choice of v4 in R^m?

Functions and graphs

Consider the functions (see attachment) and their graphs. (a) What is the largest domain, and corresponding range, of each function? (see attachment for rest of questions)

Functions and partial fractions

Show that just one of the functions (see attached file) is one-to-one on its largest domain.