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## Homework Statement

Suppose that

**A**,

**B**, and

**C**are not linearly independent. Then show how the [tex]\alpha_{i}[/tex] can be computed, up to a common factor, from the scalar products of these vectors with each other.

Hint: Suppose that there are non-zero values of the [tex]\alpha_{i}[/tex]'s that satisfy [tex]\alpha_{1}{\mathbf{A}+{\alpha_{2}{\mathbf{B}+{\alpha_{3}{\mathbf{C}}=0.[/tex] Then, taking the dot product of both sides of this equation with

**A**will yield a set of equations that can be solved for the [tex]\alpha_{i}[/tex]'s.

## Homework Equations

[tex]\alpha_{1}{\mathbf{A}+{\alpha_{2}{\mathbf{B}+{\alpha_{3}{\mathbf{C}}=0.[/tex]

## The Attempt at a Solution

Based off the instructions and hint I think they are asking me to solve for the [tex]\alpha[/tex]s of the following three equations:

[tex]\alpha_{1}{\mathbf{A{\cdot}A}+{\alpha_{2}{\mathbf{B{\cdot}A}+{\alpha_{3}{\mathbf{C{\cdot}A}}=0.[/tex]

[tex]\alpha_{1}{\mathbf{A{\cdot}B}+{\alpha_{2}{\mathbf{B{\cdot}B}+{\alpha_{3}{\mathbf{C{\cdot}B}}=0.[/tex]

[tex]\alpha_{1}{\mathbf{A{\cdot}C}+{\alpha_{2}{\mathbf{B{\cdot}C}+{\alpha_{3}{\mathbf{C{\cdot}C}}=0.[/tex]

Or, they could be asking me to solve for [tex]\alpha_{i}[/tex] of these 3 equations:

[tex]\alpha_{i}({\mathbf{A{\cdot}A}+{\mathbf{B{\cdot}A}+{\mathbf{C{\cdot}A}})=0.[/tex]

[tex]\alpha_{i}({\mathbf{A{\cdot}B}+{\mathbf{B{\cdot}B}+{\mathbf{C{\cdot}B}})=0.[/tex]

[tex]\alpha_{i}({\mathbf{A{\cdot}C}+{\mathbf{B{\cdot}C}+{\mathbf{C{\cdot}C}})=0.[/tex]

How do I solve for alpha? Matrices? Substitution? Elimination?

I'm not really sure what they mean by a common factor. Insight is always appreciated.

*Edited: Thanks for pointing that out Mark44!

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