For the following questions you may take this as a definitio

For the following questions you may take this as a definition of General Vector Space. Definition 1 (General Vector Space) A set V is a real vector space if it has a scalar multiplication and an addition operation and that satisfy the following requirements. 1. There is a distinguished constant vector 0 in V . 2. For any real number k and any vector v in V the product kv is also a vector in V . 3. For any vectors u and v in V the sum u + v is also a vector in V . 4. The following identities hold for any vectors u, v, w in V and any real numbers j,

0v = 0 (1)

1v = v

(jk)v = j(kv)

(j + k)v = jv + kv

k(u + v) = ku + kv

(u + v) + w = u + (v + w)

THE PROBLEM TO SOLVE

Indicate which of the following collections of objects for a vector space with the given operations. Here you may assume that the parent space is a subspace, so you only have to check the closure conditions to see if you have a valid subspace. You do not need to give a detailed verification for those that pass, but do give an explicit example of bad behavior for those that fail.

(a) All symmetric matrices, as part of the space M3,3 of 3 × 3 matrices with the usual operations. Yes, or No, because:

(b) All upper triangular and lower triangular matrices from M2,2, of 2 × 2 matrices. Yes, or No, because:

(c) All polynomials f(x) of degree at most 4 that satisfy f(2) = 0. Yes, or No, because:

(d) All polynomials f(x) of degree at most 4 that satisfy f(2) = 5. Yes, or No, because:

Solution

a)_

Yes. Because sum of any two symmetric matrices is also symmetric matrix

b)

YEs because sum of any two upper triangular matrices is also an upper triangular matrix

Sum of any two lower triangular matrices is alos a lower triangular matrix

c)

LEt, f ,g polynomials which satisfy

(f+g)(0)=f(0)+g(0)=0

Hence, f+g is in the set

HEnce a subspace

d)

No

LEt, f be in this set

So, f(2)=5

(2f)(2)=2f(2)=10

HEnce, 2f is not in the set ie set if not closed under scalar multiplication