Question

Let V:=Span{cost,sint}. Show that these two vectors, namely cost and sint, are linearly independent and, therefore, form a basis in V. What is the dimension of this vector space?

Answer 1

The vectors cos(t) and sin(t) are linearly independent because the only coefficients that satisfy a*cos(t) + b*sin(t) = 0 for all t are a = 0 and b = 0. They form a basis for the vector space V, which has a dimension of 2.

Step-by-step explanation:

To show that the vectors cos(t) and sin(t) are linearly independent, we must prove that the only solution to the equation acos(t) + bsin(t) = 0 for all t is a = 0 and b = 0. If we take t = 0, we get a = 0 because sin(0) = 0 and cos(0) = 1 implies that a+0=0. Similarly, by taking t = π/2, we get b = 0 since cos(π/2) = 0 and sin(π/2) = 1 yields 0+b = 0. As both coefficients a and b must be zero, the vectors are linearly independent.

Since the vectors are linearly independent and span the vector space V, they form a basis for V. The dimension of a vector space is defined by the number of vectors in its basis. Consequently, the dimension of V is 2, as there are two basis vectors, cos(t) and sin(t).