Vector Space Axioms

Closure under Addition (Axiom 1)

A + B ∈ V

Commutativity under Addition (Axiom 2)

A + B = B + A

Associativity under Addition (Axiom 3)

(A + B) + C = A + (B + C)

Addition of the zero vector (Axiom 4)

A + 0 = A (The zero matrix is the additive identity)

Negative vector under addition (Axiom 5)

A + (-A) = 0

Closure under Scalar Multiplication (Axiom 6)

c x A = cA

Distributivity of scalar multiplication (Axiom 7)

c(A + B) = cA + cB

Distributivity of scalar addition (Axiom 8)

(c + d)A = cA + dA

Associativity of scalar multiplication (Axiom 9)

c(dA) = (cd)A

Multiplicative identity (Axiom 10)

1 x A = A

Requirements for Subspace

Axioms 1, 4, and 6 are satisfied

0 vector must be in the vector space

Closure under addition

Closure under scalar multiplication

Null Space procedure

convert matrix into RREF then write result as a linear combination of vectors [x₁,x₂,x₃,x₄]^T = x₃[ ] + x₄[ ] to then form the span which will be equal to the null space

Span format

S = { (a c)^T, (b d)^T }

Linear Independence of Matrix

Partition matrix with a 0 column

If free variables after Gauss-Jordan —> Linearly Dependent

If no free variables —> Linearly Independent

Linear Independence of Polynomials

Take the Wronskian

If W = 0 —> Linearly Dependent

If W =/= 0 —> Linearly Independent

Basis process

Perform RREF, circle the pivot columns, rewrite the numbers from the original columns in span format.

Column space process

perform RREF, circle pivot columns, display as a basis (it essentially is the same as a basis)

Row space process

Write the span of your matrix in RREF but with the numbers going horizontally rather than vertically

vector b is in the column space of A if

it forms a consistent system of equations after applying Gauss-Jordan elimination

Rank of a Matrix

The number of linearly independent rows/nonzero rows in REF

All 10 Axioms in variable expression only

A + B E V

A + B = B + A

(A + B) + C = A + (B + C)

A + 0 = A where 0 E V

A + (-A) = 0

c * A = cA

c(A + B) = cA + cB

(c + d)A = cA + dA

c(dA) = (cd)A

1 * A = A

Symmetry property of Inner product

u v = v u

Additive property of Inner product

(u + w) v = u v + w * v

Homogeneity property of inner product

c(u v) = (cu) v

Positivity property of inner product

v v > 0 and v v = 0 if v = 0