I'm having some trouble with these 2 question on spanning, my prof has not given us any examples and I can't find any help from the book, any help would be awesome.
1. Let u,v be two vectors in R^n. Show that
span{u,v} = span{2u + 3v, u + 2v}
2. Let u,v,w be two vectors in R^n. Show that
span{u,v,w} = span{u + v, u + w, v + w}
"span", not "spam"!
Yet another way to show this: Any vector in the span of {2u+3v, u+2v} must be of the form a(2u+3v)+ b(u+2v)= 2au+ 3av+ bu+ 2bv= (2a+ b)u+ (3a+2b)v showing that it is in the span of {u,v}.
Conversely, any vector in the span of {u, v} must be of the form au+ bv. To have that of the form x(2u+3v)+ y(u+2v)= (2x+y)u+ (2x+2y)v we must have 2x+ 3y= a and x+ 2y= b. Solving those equations for x and y, we get x= 2a- 3b and y= 2b-a. Actually, we only have to show that we can solve those equations and that can be done by showing that the determinant of the coefficient array, and is not 0.