Question

Partial Differential Equation, first two.

Answer 1

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.