Two finitedimensional vector spaces are equal if they have t

Two finite-dimensional vector spaces are equal if they have the same operations of addition and scalar multiplication. they have a common basis. one is a subset of the other. they have the same dimension. Let B be an ordered basis for a vector space V. If the set S is formed by removing a single vecto- from B, then span(S) is not a vector space. S is lirearlv cependen:. span(S)= span(R) Sdoes not span V. Let dim(V)=n and let B and B\' be two ordered bases for V. Then, the transition matrix from B to B\' has which of the following properties? It is row equivalent to the identity matrix. The column space is all of R^n. It has a nonzero determinant. All of the above. If A is a 4 Times 4 matrix, which of the following is not a subspace of R^4? N(A) The span generated by the rows of A. The set of solutions to Ax = b for b = (1, 1, 1, 1). col(A) Find a basis for the vector space below. {ax^2 + bx + c | a - c = o} {ax^2, bx. C} {x^2,x + l} {X^2+1,x} {cx^,. Bx, c} How many vectors does a basis for M_4 Times 5 have? 9 5 20 4

Solution

Post one more question to get the remaining answers

If there are two vector space V and W, then in order for the vector space to be equal, we need to have dim(V) = dim(W), which will give the constraint that one must be the subset of the other

Hence the correct answer is Option C

2) After removing one vector from the basis set, the vectors will be linearly independent with each other but it will not be going to span the entire space

Hence the resultant S wouldn\'t be able to span V, therefore the correct answer is Option D

3) The correct answer is Option D, since transformation will have non-zero determinant since both B and B\' have non-zero determinant, the column space will span the complete space