Elementary Linear Algebra Proof

[tylerc1991]

Homework Statement

Prove or disprove the following about vectors in $\mathbb{R}^n$: If $A \cdot B = A \cdot C$ and $A \neq O,$ then $B = C.$

Homework Equations

In this example, $O$ represents the zero vector.

Let the vectors be represented as:
$A = (a_1,a_2,\dots,a_n)$
$B = (b_1,b_2,\dots,b_n)$
$C = (c_1,c_2,\dots,c_n)$

The Attempt at a Solution

$A \cdot B = A \cdot C \iff$

$\displaystyle \sum_{k=1}^{n} a_kb_k = \sum_{k=1}^{n} a_kc_k \iff$

$\displaystyle \sum_{k=1}^{n} a_kb_k - \sum_{k=1}^{n} a_kc_k = 0 \iff$

$\displaystyle \sum_{k=1}^{n} a_kb_k - a_kc_k = 0 \iff$

$\displaystyle \sum_{k=1}^{n} a_k(b_k - c_k) = 0 \iff$

So either $a_k = 0,$ or $b_k-c_k = 0$ for $k = 1,2,\dots,n.$

But since $A \neq O, \, a_k \neq 0.$

Hence, $b_k - c_k = 0 \iff b_k = c_k \iff B = C.$

It's been a while since I wrote a proof and I felt a little shaky on line 4. Thank you for your time!

 

[I like Serena]

Hi tylerc1991!

It's only the 6th and 7th line of your proof that are shaky.
If a sum of terms is zero, that does not mean that each individual term has to be zero.
Also, if at least one component of A is not zero, then A is not the zero-vector.

Let me know if you want a hint.

 

[tylerc1991]

It's only the 6th and 7th line of your proof that are shaky.
If a sum of terms is zero, that does not mean that each individual term has to be zero.
Also, if at least one component of A is not zero, then A is not the zero-vector.

This is true. I have been dallying around with a few ideas (assuming $b_k - c_k \neq 0$ and trying to arive at a contradiction), but it seems like there is always a counterexample. I am thinking I should try to break it up into cases, but I don't want to overcomplicate it. Small hint please! :shy:

EDIT: How about a counterexample to the original claim?

$A = (2,1)$
$B = (b_1,b_2)$
$C = (c_1,c_2)$

Clearly $A \neq O.$

If we try $b_1 = 1, \, c_1 = \frac{1}{2}, \, b_2 = 3, \, c_2 = 4,$ then

$A \cdot B = A \cdot C$

and $B \neq C.$

Whoops! Haha, lesson learned! Thank you for your time!

 

[I like Serena]

Take for instance n=2 and try to find a counter example.

 

[Mark44]

Here are three vectors in R²:
a = <1, 2>
b = <2, -1>
c = <-2, 1>

What is $a \cdot b$?
What is $a \cdot c$?
Is it reasonable to conclude that b = c?