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#### karush

##### Well-known member

- Jan 31, 2012

- 2,684

Are the vectors

$$v_1=x^2+1

,\quad v_2=x+2

,\quad v_3=x^2+2x$$

linearly dependent or linearly independent?

if

$$c_1(x^2+1)+c_2(x+2)+c_3(x^2+2x)=0$$

is the system

$$\begin{array}{rrrrr}

&c_1 & &c_3 = &0\\

& &c_2 &2c_3= &0\\

&c_1 &2c_2& = &0

\end{array}$$

I presume at this point observation can be made that this linear dependent

but also...

$$\left[ \begin{array}{ccc|c} 1 & 0 & 1 & 0 \\ 0 & 1 & 2 & 0 \\ 1 & 2 & 0 & 0 \end{array} \right]

\sim

\left[ \begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array} \right]$$

$$v_1=x^2+1

,\quad v_2=x+2

,\quad v_3=x^2+2x$$

linearly dependent or linearly independent?

if

$$c_1(x^2+1)+c_2(x+2)+c_3(x^2+2x)=0$$

is the system

$$\begin{array}{rrrrr}

&c_1 & &c_3 = &0\\

& &c_2 &2c_3= &0\\

&c_1 &2c_2& = &0

\end{array}$$

I presume at this point observation can be made that this linear dependent

but also...

$$\left[ \begin{array}{ccc|c} 1 & 0 & 1 & 0 \\ 0 & 1 & 2 & 0 \\ 1 & 2 & 0 & 0 \end{array} \right]

\sim

\left[ \begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array} \right]$$

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