Looks good. I'm not exactly sure what the following notation means, though:hadizainud said:v=[5t-2]=a[t+1]+b[t-1]
(1st) at+bt=5t
(2nd) a-b=-2
v=3⁄2 v_1+7⁄2 v_2
[v]_s=[■(3⁄2@7⁄2)]
is this correct?
A basis is a set of linearly independent vectors that span a vector space. In other words, any vector in the vector space can be written as a linear combination of the basis vectors. It is important because it provides a way to uniquely represent any vector in the vector space, and it simplifies calculations and proofs in linear algebra.
P1 is the set of all polynomials of degree at most 1. In other words, it is the set of all functions of the form ax + b, where a and b are real numbers.
To find [v]s with basis S={t+1,t-1} in P1: v=5t-2, we first express v in terms of the basis vectors. In this case, v = 5t-2 = 5(t+1) - 7(t-1). Therefore, [v]S = (5,-7).
Yes, a vector space can have multiple bases. This is because a basis is not unique - there can be different sets of linearly independent vectors that span the vector space.
The dimension of P1 is 2, since it has two basis vectors (t+1 and t-1) that are linearly independent and span the entire vector space.