Is there a collectionwise normal topological vector space which is not paracompact?

I can't get this clearly in my head. I have a bunch of vectors which move around over time. After every instant I wish to check by what angle(s) have the vectors moved/rotated in space with respect to an initial reference/temporary plane (static). These...

The limitations are specified on the Wiki page: Vector space model. I am interesting in tackling the first limitation which is Long documents are poorly represented because they have poor similarity values (a small scalar product and a large dimensionality...

If V is a vector space with scalars, using the definition of a vector space show that 0 scalar * v = zero vector. How do I do this? Thanks!

Show that union of two sub spaces W1 and W2 of a vector space V is a sub space of V if and only if either W1⊆W2 or W2 ⊆W1? Help me urgently

(a) Let P2(R) denote the vector space of real polynomial functions of degree less than or equal to two and let B := [p0, p1, p2] denote the natural ordered basis for P2(R) (so pi(x) = xi). Define f 2 P2(R) by f(x) = 5x2 − 2x + 3. Write f as a linear...

Let M2 denote the set of all matrices of 2 X 2. Determine if M2 is a vector space when considered with the standard addition of vectors, but with scalar multiplication given by α ⋆(a b c d) = (αa b c αd) . In case M2 fails to be a vector space with these...

Can you help me figure this out,... Let V denote the set of all m x n matrices with real entities; so V is a vector space over R (given by the book). Let F be the field of rational numbers. Is V a vector space over F with the usual definitions of matrix...

Help ,why is this not a vector space!!? Let S={ (a1,b1) ; a1,a2 are real numbers] For (a1,a2), (b1,b2) belong to S and C is a real number. Define (a1, a2) + (b1,b2)=(a1+b1, a2-b2) and c(a1, a1)=(ca1, ca2). is this a vector space? the book says its not...