# Span and Independence

Algebra Level 4

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

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

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

1. $$\{ \color{red}{\text{red}} + \color{orange}{\text{yellow}} + \color{blue}{\text{blue}},\, \color{red}{\text{red}} + 2 \color{orange}{\text{yellow}} + 4 \color{blue}{\text{blue}} \}$$
2. $$\{ \color{red}{\text{red}},\, \color{orange}{\text{yellow}},\, \color{blue}{\text{blue}} \}$$
3. $$\{ \color{red}{\text{red}} + \color{orange}{\text{yellow}},\, \color{orange}{\text{yellow}} + \color{blue}{\text{blue}} \}$$
×