Determine whether the following sets are vector spaces If no

Determine whether the following sets are vector spaces. If not, identify the requirement which is not met and give an example of that rule being broken. The set of all polynomials of degree less than or equal to 3 The set of all pairs of real numbers (x, y) such that x > y The set of all polynomials of even degree The set of all invertible 2 Times 2 matrices Determine whether or not the subsets of R^n are vector spaces. If not, show what required properties they fail to have. Is {(a,b,a-b,a+b)} a subspace of R^4 Is {(a,0,b,l,c))} a subspace of R^5 Is {(a,b,a^2,b^2)} a subspace of R^4 Determine whether the vectors in the set S span the vector space V V=R^2 S = {[0,0], [1,1]} V=R^3 S= {[1,0,0], [0,1,0], [2,3,1]} V = R^3 S= {[1,0,-1], [2,0,4], [-5,0,2], [0,0,1]}

Solution

I am solving the second question, please post other problems in the multiple questions to get the remaining answers. Thanks

Question 2:

a) The set of all the polynomials of degree less than or equal to 3

It is a vector space since it satisfies the property of additivity and scability and even the inverse of the each polynomial exists in R^3 and zero vector i.e. the zero polynomial (0x^3 + 0x^2 + 0x + 0) also exists in the R^3 polynomial space

b) The set of all pairs of real number (x,y) such that x > y

This is not a vector space, since if we multiply the c (i.e. negative real number) then we get cx < cy, which will not satisfy the property of the vector space

Hence this space is not a vector space

c) The set of all polynomials of even degree is not a vector space since it doesn\'t satisfy the addition property

Let us assume P(x) = x^2 + x and Q(x) = x^2 - x (both are 2nd degree polynomial, even degree polynomial)

P(x) - Q(x) = 2x, which is an odd degree polynomial

d) The set of all invertible matrices is not a vector space since the zero vector is not available in the set of invertible matrices

Hence we can\'t have this operation

A + 0 = A ( since 0 matrix is an non-invertible matrix that doesn\'t belong to the set)