## Independence of functions

Decide, whether the following sets of vectors are linearly independent in the space of real functions $$\mathbb R\to\mathbb R$$ (over the field $$\mathbb R$$).

• #### Variant 1

$$\{2x-1,x-2, 3x\}$$.

• #### Variant 2

$$\{x^2+2x+3, x+1, x-1\}$$.

• #### Variant 3

$$\{\sin(x), \cos(x)\}$$.

• #### Variant 4

$$\{\sin(x+1), \sin(x+2)\, \sin(x+3)\}$$.

• #### Variant 5

$$\{\ln(x),\log(2x),\log_2(x^2)\}$$.

(i.e. the natural, decadic and binary logarithm.)