Linearly independent vs dependent functions

[member 731016]

Homework Statement

Please see below

Relevant Equations

Please see below

For part(b),

My solution is,
$(c_1, c_2, c_3, c_4) = (c_1, 5c_1, -c_1, -3c_1)$

They have taken the case that c_1 = -1, which gives their expression for a linear dependent function as they have shown. However, I'm confused that the functions are linearly dependent for any value of $c_1 \in \mathbb{R}$. Since if we take $c_1 = 0$, then $c_1 = c_2 = c_3 = c_4 = 0$. I agree thought they the functions are linear dependent for $c_1 \in \mathbb{R}\{0\}$ (Set of real numbers without zero)

Does someone please know why they did not include that case?

Thanks!

 

[docnet]

Homework Statement: Please see below
Relevant Equations: Please see below

For part(b),
View attachment 346379
My solution is,
$(c_1, c_2, c_3, c_4) = (c_1, 5c_1, -c_1, -3c_1)$

They have taken the case that c_1 = -1, which gives their expression for a linear dependent function as they have shown. However, I'm confused that the functions are linearly dependent for any value of $c_1 \in \mathbb{R}$. Since if we take $c_1 = 0$, then $c_1 = c_2 = c_3 = c_4 = 0$. I agree thought they the functions are linear dependent for $c_1 \in \mathbb{R}\{0\}$ (Set of real numbers without zero)

Does someone please know why they did not include that case?

Thanks!

Heya! From the definition of linear dependence, the functions
$$f_1,f_2,...,f_k$$ are linearly dependent if there exist scalars $$a_1,a_2,..,a_k,$$ not all zero, such that
$$a_1f_1+\cdots f_ka_k=0.$$

 

[Orodruin]

They have taken the case that c_1 = -1, which gives their expression for a linear dependent function as they have shown. However, I'm confused that the functions are linearly dependent for any value of $c_1 \in \mathbb{R}$. Since if we take $c_1 = 0$, then $c_1 = c_2 = c_3 = c_4 = 0$. I agree thought they the functions are linear dependent for $c_1 \in \mathbb{R}\{0\}$ (Set of real numbers without zero)

That is not at all what they are saying. Being linearly dependent means that there is a non-zero solution to
$$
\sum_{i = 1}^4 c_i f_i(t) = 0
$$
and their solution has all $c_i \neq 0$. It makes no sense to say that the functions are linearly independent for some particular values of $c_i$. Being linearly independent just means that there exists a non-zero solution for which the above equation holds for all $t \in \mathbb R$ (not $c_i \in \mathbb R$), which is what they are saying.

Edit: Note: The correct $\LaTeX$ for the set you are referring to (the reals apart from zero) is $\mathbb R \setminus \{0\}$.

 

[WWGD]

If $c1f_1+c_2f_2+ c_3 f_3+ c_4f4=0$, then $-( c_1f_1+ c2f_2+ c_3f_3+c_4 f_4)=0$. Notice each of your terms is,the negative of those in the solution from the book.