Chapter 3: Vector Spaces and Subspaces Flashcards
Subspace
A subspace of a vector is a set of vectors (including 0) hat satisfies requirements:
- v+w=subspace
- cv =subspace
- cv + dw =subspace
Column Space of A
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.
Nullspace
&
Special Solutions
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)
Pivot Columns and Free Columns
After reducing matrix to RREF (Reduced Row Echelon Form):
Columns with pivots are Pivot Columns
Columns with no pivots are Free Columns
Rank
Rank = r = number of pivots
If r = m = n
The matrix is square (m = n)
The matrix is invertible (r !< n)
Ax = b has one solution
Augmented Matrix
[A | b]
Full Column Rank
- All columns have pivots
- There are no free variables or special solutions
- The nullspace of A contains only the zero vector
Ax=b only has one solution
Full Row Rank
- All the rows have pivots
- Ax=b has a solution for every in Ax=b
- The col space is R^m
- n - r = n - m special solutions
r=m r
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
r
The matrix is tall/thin
ax=b has 0 or 1 solution
r
Not full rank
Axb has 0 or infinite solutions
A is singular/not-invertible
Linear Independence
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
Span
Vectors span a subspace if their linear combinations fill the space
Four Fundamental Subspaces
- The rowspace is C(A’) a subspace of R^n
- The column space is C(A) a subspace of R^m
- The nullspace is N(A) a subspace of R^n
- 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
Fundamental Theorem of Linear Algebra (P1)
The column space and the row space both have dimension r.
Nullspace has dimensions n-r and m-r.
The 8 Axioms of Vector Spaces
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
RREF
Reduced Row Echelon Form
The result of full Gauss-Jordan elimination
