**1. The problem statement, all variables and given/known data**

"Simplify: (a+2b+3c, 4a+5b+6c, 7a+8b+9c)."

All of these are vectors ([tex]a=\vec{a}[/tex], etc.)

**2. Relevant equations**

I know that set of vectors [tex]\vec{x}_1+\cdots+\vec{x}_n \in V[/tex] is linearly independent

if linear combination [tex]\alpha_1\vec{x}_1+\cdots+\alpha_n\vec{x}_n[/tex] equals [tex]\vec{0}[/tex]

iff all scalars [tex]\alpha_1\,\cdots,\alpha_n[/tex] equal 0.

**3. The attempt at a solution**

(I don't understand it, although it's (partially) in my notebook)

(a+2b+3c, 4a+5b+6c, 7a+8b+9c) =

= (a,4a,7a) + (a,4a,8b) + (a,4a,9c) +

+ (a,5b,7a) + (a,5b,8b) + (a,5b,7a) +

+ (a,6c,8b) + ____0___ + ___0____ +

+ (2b,4a,9c) + (2b,6c,7a) +

+ (3c,4a,8b) + (3c,5b,7a) + =

= 48(a,b,c) 72(a,b,c) + 84(a,b,c) + 128(a,b,c) - 105(a,b,c) + 45(a,b,c)

= (a,b,c)(45+48-72+84+128)

The whole first 2 lines

*" = (a,4a,7a) + (a,4a,8b) + (a,4a,9c) +*

+ (a,5b,7a) + (a,5b,8b) + (a,5b,7a) + "

+ (a,5b,7a) + (a,5b,8b) + (a,5b,7a) + "

were removed, because these vectors were lin. dependent.

So, only vectors as [tex](\alpha\vec{a}, \beta\vec{b}, \gamma\vec{c})[/tex] count as lin.

*in*dependent. But, if you add some of the last "simplified" ones

*...+(2b,4a,9c) + (2b,6c,7a) + (3c,4a,8b) + (3c,5b,7a)*and

**add them, you don't get the original one?!**

**And, how is (a,6c,8b) same as 48(a,b,c)? Or where did**

*48*come from?Addendum: How do you make strikethrough block of text? I've tried with HTML's <s> and <strike> and <del>, also found nothing with [tex]LaTeX[/tex].