# Is $f_1$, $f_2$ and $f_3$ are linearly independent?

Let $\mathcal{C}(\mathbb{R})$ be the $\mathbb{R}$-vector space of continuous functions from $\mathbb{R}$ to $\mathbb{R}$. Let $a_1$, $a_2$, $a_3$ be distinct real numbers. For $i = 1, 2, 3$, let $f_i \in \mathcal{C}(\mathbb{R})$ be the function $f_i(t)=e^{a_it}$ . Which of the following statement(s) is/are true?

(A) $f_1$, $f_2$ and $f_3$ are linearly independent

(B) $f_1$, $f_2$ and $f_3$ are linearly dependent.

(C) $f_1$, $f_2$ and $f_3$ form a basis of $\mathcal{C}(\mathbb{R})$.

My attempt: my answer is option A and C, because if we check the linearly independent that is,,,,, $c_1f_1 + c_2f_2 + c_3f_3 =0$ , $f_i(t)=e^{a_it} \neq 0$ where $c_1= c_2 = c_3= 0$,,,so $f_1$, $f_2$ and $f_3$ will form a basis of $\mathcal{C}(\mathbb{R})$ that mean $f_1$, $f_2$ and $f_3$ are linearly independent..so correct option is opton A and option C,,,

Is my answer is correct or incorrect. pliz verified my answer and tell me the solution,

• Consult this link. Oct 28, 2017 at 9:06

No, the correct answer is A. They don't for a basis of $\mathcal{C}(\mathbb{R})$. For instance, you cannot express the identity as a linear combination of your functions.