Partial Differential Equation, first two.

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Partial Differential Equation, first two.

Mathematics
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Question 2 is fairly straight forward. Assuming, of course, that all the vectors \[Y_{1}, Y _{2}, Y _{3}\] are non-zero, let's see what happens if \[k _{3}\]is non-zero. We can rearrange the equation \[k _{1}Y _{1}+k _{2}Y _{2}+k _{3}Y _{3} = 0\] into \[Y _{3} =1/k _{3}(-k _{1}Y _{1}-k _{2}Y _{2})\] so that the right side is non-zero because the left is non-zero, and it is defined because \[k _{3}\neq0.\] Note that \[Y _{3}\] is now a linear combination of the other two vectors, or is in \[Span \left\{ Y _{1} ,Y _{2}\right\}\] if you prefer. Now do the same for the other two vectors.

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