Question 1.11.5: Show that the set of all linear combinations of the sine and...

We let C denote the vector space of all continuous functions, and now let H be the subset of C which is defined to be all functions that are linear combinations of sin t and cos t . That is, a typical element of H is a function f of the form

f (t ) = c_{1} sin t +c_{2} cos t

where c_{1} and c_{2} are any real scalars. We need to show that the set H contains the zero function from C, that H is closed under scalar multiplication, and that H is closed under addition.

First, if we choose c_{1}=c_{2}=0, the function z(t )=0sin t +0cos t =0 is the function that is identically zero, which is the (continuous) zero function from C. Next, if we take a function from H, say f (t ) = c_{1} sin t +c_{2} cos t , and multiply it by a scalar k, we get

kf (t ) = k(c_{1} sin t +c_{2} cos t ) = (kc_{1}) sin t +(kc_{2}) cos t

which is of course another element in H, so H is closed under scalar multiplication. Finally, if we consider two elements f and g in H, given by f (t ) = c_{1} sin t +c_{2} cos t and g (t ) = d_{1} sin t +d_{2} cos t , then it follows that

f (t )+g (t ) = (c_{1} sin t +c_{2} cos t )+(d_{1} sin t +d_{2} cos t)
= (c_{1} +d_{1}) sin t +(c_{2} +d_{2}) cos t

so that H is closed under addition, too. Thus, H is a subspace of C.