Hello, i have a true false question about vector spaces that i cannot figure out and its part of our test review so i need to know it.
A.)The columns of an invertible nxn matrix form a basis for R^n
B.)In some cases, the linear dependence relations amoung the columns of a matrix can be affected by certain elementary row operations on the matrix.
C.)A single vector by itself is linearly dependent
D.)if H=Span(b1,.....,bp) then (b1,.....,bp) is a basis for H
E.)A basis is a spanning set that is as large as possible.
I think that A,B,C are true. Am i right? and are the others true?
Well (C) is false, unless the vector is the 0 vector (why?).
For (B), I want to say that that is also false (i guess it depends what "affect" means).
D and E are false as well, both with simple explanations. What happens if you repeat a vector in a spanning set? What is the largest spanning set you could possibly take?