Chapter 3: Vector Spaces and Subspaces

Question 1

Subspace

Answer 1

A subspace of a vector is a set of vectors (including 0) hat satisfies requirements:

1. v+w=subspace
2. cv =subspace
3. cv + dw =subspace

Question 2

Column Space of A

Answer 2

The col space consist of ALL LINEAR COMBINATIONS OF THE COLUMNS
(all possible valid solutions ‘b’ for Ax=b)
The combinations are all possible vectors A(x) that fill C(A)
One easy basis for C(A) is the space described by only the pivot columns of A.

Question 3

Nullspace
&
Special Solutions

Answer 3

Vectors in R^n that contain all solutions to AX=0
Nullspace basis vectors are the vectors of the special solutions to Ax=0
Special solutions are found by solving the systems of equations presented by R(A) = 0.
The number of special solutions correspond to the number of free columns (s = m - r)

Question 4

Pivot Columns and Free Columns

Answer 4

After reducing matrix to RREF (Reduced Row Echelon Form):
Columns with pivots are Pivot Columns
Columns with no pivots are Free Columns

Question 5

Rank

Answer 5

Rank = r = number of pivots

Question 6

If r = m = n

Answer 6

The matrix is square (m = n)
The matrix is invertible (r !< n)
Ax = b has one solution

Question 7

Augmented Matrix

Answer 7

[A | b]

Question 8

Full Column Rank

Answer 8

1. All columns have pivots
2. There are no free variables or special solutions
3. The nullspace of A contains only the zero vector
   Ax=b only has one solution

Question 9

Full Row Rank

Answer 9

1. All the rows have pivots
2. Ax=b has a solution for every in Ax=b
3. The col space is R^m
4. n - r = n - m special solutions

Question 10

r=m r

Answer 10

The matrix is short/wide
Ax=b has infinite solutions
Nullspace is at least one-dimensional with at least one special solution.
(n - r) special solutions
(n - r) nullspace dimensions

Question 11

r

Answer 11

The matrix is tall/thin

ax=b has 0 or 1 solution

Question 12

r

Answer 12

Not full rank
Axb has 0 or infinite solutions
A is singular/not-invertible

Question 13

Linear Independence

Answer 13

Property of a matrix, not individual vectors
Columns of A are linearly independent when the only solution to Ax=0 is x=0
x1v1+x2v2+…xnvn=0

Question 14

Span

Answer 14

Vectors span a subspace if their linear combinations fill the space

Question 15

Four Fundamental Subspaces

Answer 15

1. The rowspace is C(A’) a subspace of R^n
2. The column space is C(A) a subspace of R^m
3. The nullspace is N(A) a subspace of R^n
4. The left nullspace is N(A’) a subspace of R^m

Dimensions:
Column number (n) match rowspace and nullspace. C(A'),N(A) are in R^n
Row number (m) matches column space and left-nullspace. C(A),N(A') are in R^m

Question 16

Fundamental Theorem of Linear Algebra (P1)

Answer 16

The column space and the row space both have dimension r.

Nullspace has dimensions n-r and m-r.

Question 17

The 8 Axioms of Vector Spaces

Answer 17

To qualify as a vector space, the set V and the operations of addition and multiplication must adhere to a number of requirements called axioms.In the list below, let u, v and w be arbitrary vectors in V, and a and b scalars in F.
1. Associativity of addition
u + (v + w) = (u + v) + w
2. Commutativity of addition
u + v = v + u
3. Identity element of addition
There exists an element 0 ∈ V, called the zero vector, such that v + 0 = v for all v ∈ V.
4. Inverse elements of addition
For every v ∈ V, there exists an element −v ∈ V, called the additive inverse of v, such that v + (−v) = 0.
5. Compatibility of scalar multiplication with field multiplication
a(bv) = (ab)v
6. Identity element of scalar multiplication
1v = v, where 1 denotes the multiplicative identity in F.
7. Distributivity of scalar multiplication with respect to vector addition
   a(u + v) = au + av
8. Distributivity of scalar multiplication with respect to field addition
(a + b)v = av + bv

Question 18

RREF

Answer 18

Reduced Row Echelon Form

The result of full Gauss-Jordan elimination