Question

if c.f(x) + k.g(x) = 0 -----(1)
if we can find non-zero constants c and k for which (1) will also be true for all x then we call the two functions linearly dependent. On the other hand if the only two constants for which (1) is true are c = 0 and k = 0 then we call the functions linearly independent. It's Wronskian's application. My question is that what if only one constant is zero & another is non-zero then what will we call it?

Answer 1

suppose c is not zero but k = 0 and (1) is satisfied?

Then clearly f is the zero function.

Answer 2

will the equation (1) remain linearly dependent or linearly independent or semi linearly independent or semi linearly dependent

Answer 3

I mean the function

Answer 4

If a vector is zero (or a function is zero in the vector space of functions), we do not say it is linearly indep or dependent with respect to another non-zero vector.

For example, in \( \mathbb{R}^2 \), the vectors (0,0) and (0,1) are not linearly dependent or linearly independent.

Answer 5

not the equation

Answer 6

hyper-hyper-technically, you could say that (0,0) and (0,1) are linearly independent, but we never think of it that way.

Answer 7

thanks

Answer 8

the answer to your question:
"will the equation (1) remain linearly dependent or linearly independent or semi linearly independent or semi linearly dependent"
is none of the above. We do not consider the zero vector as being lin. indep or dep.

Answer 9

thanks