Vector Space Axioms

0.0(0)
learnLearn
examPractice Test
spaced repetitionSpaced Repetition
heart puzzleMatch
flashcardsFlashcards
Card Sorting

1/20

encourage image

There's no tags or description

Looks like no tags are added yet.

Study Analytics
Name
Mastery
Learn
Test
Matching
Spaced

No study sessions yet.

21 Terms

1
New cards

Closure under Addition (Axiom 1)

A + B ∈ V

2
New cards

Commutativity under Addition (Axiom 2)

A + B = B + A

3
New cards

Associativity under Addition (Axiom 3)

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

4
New cards

Addition of the zero vector (Axiom 4)

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

5
New cards

Negative vector under addition (Axiom 5)

A + (-A) = 0

6
New cards

Closure under Scalar Multiplication (Axiom 6)

c x A = cA

7
New cards

Distributivity of scalar multiplication (Axiom 7)

c(A + B) = cA + cB

8
New cards

Distributivity of scalar addition (Axiom 8)

(c + d)A = cA + dA

9
New cards

Associativity of scalar multiplication (Axiom 9)

c(dA) = (cd)A

10
New cards

Multiplicative identity (Axiom 10)

1 x A = A

11
New cards

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

12
New cards

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

13
New cards

Span format

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

14
New cards

Linear Independence of Matrix

Partition matrix with a 0 column

If free variables after Gauss-Jordan —> Linearly Dependent

If no free variables —> Linearly Independent

15
New cards

Linear Independence of Polynomials

Take the Wronskian

If W = 0 —> Linearly Dependent

If W =/= 0 —> Linearly Independent

16
New cards

Basis process

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

17
New cards

Column space process

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

18
New cards

Row space process

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

19
New cards

vector b is in the column space of A if

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

20
New cards

Rank of a Matrix

The number of linearly independent rows/nonzero rows in REF

21
New cards

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