Answer:
Step-by-step explanation:
STEP 1 It's true
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.