Question

Consider the following statements about the linear dependence of the real valued functions $y_1=1,y_2=x$ and $y_3=x^2$, over the field of real numbers.

I. $y_1,y_2$ and $y_3$ are linearly independent on $-1\le x\le 0$

II. $y_1,y_2$ and $y_3$ are linearly dependent on $0\le x\le 1$

III. $y_1,y_2$ and $y_3$ are linearly dependent on $0\le x\le 1$

IV. $y_1,y_2$ and $y_3$ are linearly independent on $-1\le x\le 1$

Which on among the following is correct?

• A. Both I and IV are True
• B. Both I and III are True
• C. Both II and IV are True
• D. Both III and IV are True

Answer 1

The correct option is A Both I and IV are True

$y_1=1,y_2=x,y_3=x^2$

Linear combination is given by

$a{y}_1+b{y}_2+c{y}_3=0,a,b,cϵR$

$\Rightarrow a+bx+c{x}^2=0a,b,c,ϵR$

Case : 1 If $xϵ[0,1]$

a +bx +cx${}^2$ = 0

At x = 0 $\Rightarrow$ a = 0 ....(1)

$Atx=\frac{1}{2}\Rightarrow \frac{b}{2}+\frac{c}{4}=0$ ...(2)

At x = 1, b + c = 0

From equation (1) and (2), we get

b = c = 0

$\therefore$ a = b = c = 0

$\Rightarrow y_1,y_{2}and{y}_3$ are linearly independent for $0\le x\le 1$

Case II: If $xϵ[-1,0]$

a + bx +$c{x}^2=0$

At x = 0 $\Rightarrow a=0$

At x = -1 $\Rightarrow -b+c=0...\left(4\right)$

$Atx=-\frac{1}{2}\Rightarrow -\frac{b}{2}+\frac{c}{4}=0....\left(5\right)$

From equation (4) and (5), we get

b = c = 0

a = b = c = 0

$y_1,y_2$ and $y_3$ are linearly independent on $-1\le x\le 0$