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. \(\{ \color{red}{\text{red}} + \color{orange}{\text{yellow}},\, \color{orange}{\text{yellow}} + \color{blue}{\text{blue}} \}\)
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