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

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In summary, the conversation discusses simplifying a set of vectors (a+2b+3c, 4a+5b+6c, 7a+8b+9c) by representing them as a linear combination of linearly independent vectors (\vec{a}, \vec{b}, \vec{c}). The attempt at a solution involves removing some of the vectors that are linearly dependent and rearranging the remaining vectors to form a linear combination. However, the final result does not match the original set of vectors, leading to confusion about the concept of linear independence. The conversation also mentions the use of HTML and LaTeX tags for formatting math equations.
  • #1
courteous
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Homework Statement


"Simplify: (a+2b+3c, 4a+5b+6c, 7a+8b+9c)."
All of these are vectors ([tex]a=\vec{a}[/tex], etc.)

Homework 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.

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) + "

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. independent. 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].
 
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  • #2
What do you mean by "simplified"? What I would do is write it as a "linear combination" : (a+2b+3c, 4a+5b+6c, 7a+8b+9c)= a(1, 4, 7)+ b(2, 5, 8)+ c(3, 6, 9).
 
  • #3
Instructions say (word-for-word):
Let [tex]\vec{a},\vec{b},\vec{c}[/tex] be linearly independent. Simplify: [tex](a+2b+3c, 4a+5b+6c, 7a+8b+9c)[/tex].

PS: Where's the button for tex tags?
 

What does it mean to "simplify" an expression?

To simplify an expression means to reduce it to its most basic form by combining like terms and eliminating any unnecessary elements.

How do I simplify an expression with multiple variables?

To simplify an expression with multiple variables, first combine all like terms by adding or subtracting their coefficients. Then, arrange the terms in alphabetical order according to the variables. If there are any remaining like terms, combine them as well.

What is the purpose of simplifying an expression?

The purpose of simplifying an expression is to make it easier to work with and to better understand its underlying structure. It also allows for more efficient calculations and problem solving.

Can I simplify an expression with different types of variables?

Yes, you can simplify an expression with different types of variables as long as they are like terms. For example, you can combine terms with x and terms with y, but you cannot combine terms with x and terms with z.

Is there a specific order to simplify expressions with multiple terms and variables?

Yes, the typical order of simplification is to first combine like terms, then arrange the terms in alphabetical order according to the variables. However, in some cases, it may be more efficient to rearrange the terms before combining them. Ultimately, the order may vary depending on the specific expression and the desired result.

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