## anonymous 5 years ago Partial Differential Equation, first two.

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.