Question: Let W be the space spanned by f = sinx and g = cosx.
(a) Show that for any value of theta, f, = sin(x + theta) and g_1 = cos (x + theta) vectors in W.
(b) Show that f_1 and g_1 form a basis for W.© BrainMass Inc. brainmass.com March 4, 2021, 6:53 pm ad1c9bdddf
Because f = sin x and g = cos x span the vector space S, any linear or multiplicative combination of these two vectors will also be vectors in W.
Consider u=sin(x+theta) and v=cos(x+theta)
By trigonometric identities,
Then, u = sinx ...
In about 180 words, this solution discusses the two proofs regarding vector spaces. The solution is brief, but concise and will be understandable for anyone with a basic understanding of this subject material.