Exercise 5

Which of these subsets of the space of real-valued functions of one real variable is linearly dependent and which is linearly independent? (We have abbreviated some constant functions; e.g., in the first item, the ‘$2$’ stands for the constant function $f(x)=2$.)

• (a)  $\{\hskip 1.00006pt2,4\sin^{2}(x),\cos^{2}(x)\hskip 1.00006pt\}$     (b)  $\{\hskip 1.00006pt1,\sin(x),\sin(2x)\hskip 1.00006pt\}$     (c)  $\{\hskip 1.00006ptx,\cos(x)\hskip 1.00006pt\}$     (d)  $\{\hskip 1.00006pt(1+x)^{2},x^{2}+2x,3\hskip 1.00006pt\}$     (e)  $\{\hskip 1.00006pt\cos(2x),\sin^{2}(x),\cos^{2}(x)\hskip 1.00006pt\}$     (f)  $\{\hskip 1.00006pt0,x,x^{2}\hskip 1.00006pt\}$