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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