Vector Spaces Associated to Matrices

Question 1

Define column space.

Answer 1

Let A be a m × n matrix. Then, the Column Space (AKA the Image of A – Im(A)) is Col(A) := Span{c1, · · · , cn},
where c1, · · · , cn are m × 1 columns of A (or vectors in R^m).
Recall from matrix multiplication that if x ∈ R^n, then Ax is a linear combination of the columns of A. So, Col(A) = {Ax | x∈R^n}.
Col(A) is a subspace of R^m.

Question 2

Define row space.

Answer 2

The Row Space of a m × n matrix A is
Row(A) = Span{r1, · · · ,rm},
where r1, · · · ,rm are 1 × n rows of A (or vectors in R^n). Therefore, Row(A) is a subspace of R^n.

Question 3

What is another term for nullspace?

Answer 3

Kernel

Question 4

Define nullspace.

Answer 4

The nullspace of A (AKA as the kernel of A – ker(A)) is
ker(A) = {x ∈ R^n | Ax = 0}, or
The kernel is the general solution to the homogeneous linear system Ax = 0. ker(A) is a subspace of R^n.

Question 5

Explain how basic solutions form a basis of kernel.

Answer 5

The spanning set of ker(A) obtained from the RREF of [A|0] (or the set of basic solutions of Ax = 0) is a basis for ker(A).

Question 6

Define rank-nullity theorem.

Answer 6

Let A be a m × n matrix. The dimension of ker(A) is equal to the number of non-leading variables in [A|0], or
dim (ker(A)) + rank(A) = n (or # columns of A)

Question 7

Describe the relationship between solving inhomogeneous systems and kernel.

Answer 7

Suppose Ax = b is a consistent linear system. Then,
1. If x=v is a solution to Ax=b and x=u is any
solution to Ax=0, then u+v is a solution to Ax=b.
2. If v and w are two particular solutions to Ax=b, then x=w−v is a solution to Ax = 0, or x=w−v belongs to ker(A).

Question 8

What are 3 facts of linear combinations relevant here?

Answer 8

1. If Ax = 0 has a unique solution, so does any consistent Ax = b,
2. If Ax = 0 has infinitely many solutions with k parameters in the general solution, then any consistent Ax = b will also have infinitely many solutions with k parameters in the general solution,
3. If b ∈/ Col(A), then Ax = b is inconsistent.

Question 9

Give a Summary of Consistency of Linear Systems

Answer 9

Suppose A is a m × n matrix. The linear system Ax = b is consistent if and only if

1. b is a linear combination of the columns of A,
2. or b ∈ Col(A),
3. or rank(A) = rank([A|b])

Question 10

The system Ax = b is consistent for all b ∈ R

m if and only if: (5)

Answer 10

1. Every b ∈ R^m is a linear combination of the columns of A, or
2. there are no zero rows in the RREF of A, or
3. Col(A) = R^m, or
4. dim(Col(A)) = m, or
5. rank(A) = m.

Question 11

A consistent system Ax = b has a unique solution ⇔ (7)

Answer 11

1. Every variable is a leading variable, or
2. there is leading 1 in every column of the RREF of A, or
3. Ax = 0 has a unique solution, or
4. the columns of A are Linearly Independent, or
5. ker(A) = {0}, or
6. dim(ker(A)) = 0, or
7. rank(A) = n.

Question 12

What are 2 ways to describe a subspace W of R^n?

Answer 12

1. W = ker(A), for some m×n matrix A,

2. W = Span{u1, · · · , um}.