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## Homework Statement

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

## Homework Equations

In this example, [itex]O[/itex] represents the zero vector.

Let the vectors be represented as:

[itex]A = (a_1,a_2,\dots,a_n)[/itex]

[itex]B = (b_1,b_2,\dots,b_n)[/itex]

[itex]C = (c_1,c_2,\dots,c_n)[/itex]

## The Attempt at a Solution

[itex] A \cdot B = A \cdot C \iff[/itex]

[itex] \displaystyle \sum_{k=1}^{n} a_kb_k = \sum_{k=1}^{n} a_kc_k \iff[/itex]

[itex] \displaystyle \sum_{k=1}^{n} a_kb_k - \sum_{k=1}^{n} a_kc_k = 0 \iff [/itex]

[itex] \displaystyle \sum_{k=1}^{n} a_kb_k - a_kc_k = 0 \iff[/itex]

[itex] \displaystyle \sum_{k=1}^{n} a_k(b_k - c_k) = 0 \iff[/itex]

So either [itex] a_k = 0, [/itex] or [itex] b_k-c_k = 0 [/itex] for [itex] k = 1,2,\dots,n. [/itex]

But since [itex] A \neq O, \, a_k \neq 0.[/itex]

Hence, [itex] b_k - c_k = 0 \iff b_k = c_k \iff B = C.[/itex]

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