# Span and Independence

Algebra Level 4

The vector space of subtractive color mixing is created from real multiples of vectors $\color{#D61F06}{\text{red}}$, $\color{#EC7300}{\text{yellow}}$, and $\color{#3D99F6}{\text{blue}}$ with the understanding that $\color{#D61F06}{\text{red}} + \color{#EC7300}{\text{yellow}} + \color{#3D99F6}{\text{blue}} = \textbf{0}$. (When mixing colors, red, yellow, and blue combine to make black.)

For instance, some elements of the vector space are $\sqrt{2} \color{#D61F06}{\text{red}} + \pi \color{#EC7300}{\text{yellow}}\ \text{ and } -3 \color{#D61F06}{\text{red}} + 29 \color{#EC7300}{\text{yellow}} + e^3 \color{#3D99F6}{\text{blue}}.$ However, $3 \color{#D61F06}{\text{red}} + 2 \color{#3D99F6}{\text{blue}} \text{ and } \color{#D61F06}{\text{red}} - 2 \color{#EC7300}{\text{yellow}} \text{ are the same vector},$ since they differ by a multiple of $\textbf{0}$. $($They differ by $2 \cdot ({\color{#D61F06}{\text{red}} + \color{#EC7300}{\text{yellow}} + \color{#3D99F6}{\text{blue}}) = \textbf{0}.})$

What is true of the following sets of vectors in this vector space?

1. $\{ \color{#D61F06}{\text{red}} + \color{#EC7300}{\text{yellow}} + \color{#3D99F6}{\text{blue}},\, \color{#D61F06}{\text{red}} + 2 \color{#EC7300}{\text{yellow}} + 4 \color{#3D99F6}{\text{blue}} \}$
2. $\{ \color{#D61F06}{\text{red}},\, \color{#EC7300}{\text{yellow}},\, \color{#3D99F6}{\text{blue}} \}$
3. $\{ \color{#D61F06}{\text{red}} + \color{#EC7300}{\text{yellow}},\, \color{#EC7300}{\text{yellow}} + \color{#3D99F6}{\text{blue}} \}$
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