Span and Independence

Algebra Level 4

The vector space of subtractive color mixing is created from real multiples of vectors red\color{#D61F06}{\text{red}}, yellow\color{#EC7300}{\text{yellow}}, and blue\color{#3D99F6}{\text{blue}} with the understanding that red+yellow+blue=0\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 2red+πyellow  and 3red+29yellow+e3blue.\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, 3red+2blue and red2yellow are the same vector,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 0\textbf{0}. ((They differ by 2(red+yellow+blue)=0.)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. {red+yellow+blue,red+2yellow+4blue}\{ \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. {red,yellow,blue}\{ \color{#D61F06}{\text{red}},\, \color{#EC7300}{\text{yellow}},\, \color{#3D99F6}{\text{blue}} \}
  3. {red+yellow,yellow+blue}\{ \color{#D61F06}{\text{red}} + \color{#EC7300}{\text{yellow}},\, \color{#EC7300}{\text{yellow}} + \color{#3D99F6}{\text{blue}} \}
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