1. The problem statement, all variables and given/known data Just like to verify these statements as being always true or false since I've been told subspaces is the most important concept in matrix/linear algebra. a. set of vectors in R3 that satisfy x = |y| form a vector space of R3 b. if S is a subspace of Rn and the dimension of S = n, then S = Rn c. dimensions of Col A and Nul A add up to the number of columns of A d. if a set of p vectors spans an x-dimensional subspace C of Rn, then these vectors form a basis for C e. The dimension of Nul A is the number of variables in the equation Ax = 0 f. if C is an x-dimensional subspace of Rn, then a linearly independent set of x vectors in C is a basis for C 3. The attempt at a solution a. this is not always true because the vectors might not contain the zero vector (not going through the origin), which is required for a vector subspace. b. always true since S defines the basis so dimension of each vector in subspace is n c. false, I know that Rank of matrix + nulity A = # columns of A so number of columns = dimension of col(A) - dimension of nulity of A d. think this is false, because span is linearly dependent, but a basis needs to have linearly independent vectors. e. I think this one is always true by definition. f. This one seems tricky, so I looked up some definitions. Since there's a subspace, I can pick out any linearly independent vectors from that and so it should always be a basis for C. So i think this is true. I desperately want to understand this stuff fully with the right reasons. Any help is greatly appreciated.