If B is a basis for a subspace H, then each vector in H can be written in only one way as a linear combination of the vectors in B. The dimension of Nul A is the number of variables in the equation Ax = 0. The dimension of the column space of A is rank A. If B = {vi,...,Vp} is a basis for a subspace H of Rn, then the correspondence makes H look and act the same as Rp. If H is a p-dimensional subspace of Rn, then a linearly independent set of p vectors in H is a basis for H.

If H is a p-dimensional subspace of Rn, then B must be a set of p elements {v1, v2, . . . , vp}. To represent any vector b in H is to ﬁnd the coeﬃcients cj in the linear combination

b = c1v1 + c2v2 + . . . + cpvp

of vectors in B whose sum is b. This is equivalent to solving a system of equations with p equations and p unknowns. Because the vectors are linearly independent by the deﬁnition of a basis, this means there can only be one solution.

STEP 2 It's true

The dimension of Nul A is the number of free variables in the equation Ax = 0.

STEP 3 It's true

The rank of a matrix A is equal to the dimension of Col(A).

STEP 4 It's true

Because H is a p-dimensional subspace of Rn, any linearly independent set of exactly p elements in H is automatically a basis for H, hence also spans H

STEP 5 It's true

This correspondence is a one-to-one correspondence between H and Rp that preserves linear combinations, this is also known as an isomorphism. Because H is isomorphic to Rp, H looks and acts the same as Rp even though the vectors in H themselves may have more than p entries.