basis, rank, and dimension

a basis for a subspace of V of R^n is

a set of vectors {v1,…,vp} such that

1. the set of vectors is linearly independent

2. span {v1,..vp} = V

Basis for Col A

pivot columns of the original matrix A

Basis for Nul A

write the solution set for Ax = 0 in parametric vector form

Rank Theorem

If A is an mxn matrix, then

dim(Col A) + dim(Nul A) = n where

• dim(Col A) = # pivots in A

• dim(Nul A) = # columns w/o pivots

• n = # columns in A

Rank Nullity Theorem

rank(A) + nullity(A) = # columns in A where

• rank(A) = dim(ColA)

• nullity(A) = dim(Nul A)

Basis Theorem

suppose V is a subspace and you know dim(V) = m

1. any m linearly independent vectors in V forms a basis of V

2. any m vectors that span V form a basis for V

True or False: The set of all solutions of a system of m homogenous equations in n unknowns is a subspace of R^n

True. We can solve this system as an mxn matrix and solve matrix equation Ax = 0. The solution set is the null space of A, which is a subspace of R^n.

True or False: If B is an echelon form of a matrix A, then the pivot columns of B form a basis for the column space of A.

False. A basis of the column space of A consists of the columns of A that correspond to the pivot columns in B

True or False: The column space of an mxn matrix is a subspace of R^m

True. The column space lives in R^m and is a subspace of R^m.

True or False: Any set of n linearly independent vectors in R^n is a basis for R^n

True. Since R^n has dimension n, we know from the Basis Theorem that any set of n linearly independent vectors in R^n will form a basis of R^n.

True or False: The null space of an mxn matrix is a subspace of R^m

False, The null space lives in R^n and is a subspace of R^n.