1. ## True/False about Vector Spaces

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?

Thank You,
Diggidy

2. Originally Posted by Diggidy
I think that A,B,C are true. Am i right?
More important, show some work. Why do you think they are true?. For example: $\displaystyle A$ invertible implies $\displaystyle \det A\neq 0$ wich implies $\displaystyle \textrm{rank}A=n$ etc.

and are the others true?

A little help: for D) choose $\displaystyle H=\mathbb{R}^2$ and $\displaystyle b_1=(1,0),b_2=(2,0),b_3=(0,1)$ .

3. 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?