matrix theory exam 2

trivial solution

x = 0

nontrivial solution

x ≠ 0

only has trivial solution

linearly independent and no free variable

has nontrivial solution

linearly dependent and free variabel present

Span ℝⁿ if

-there is a pivot position in every row

-every b is a linear combination of the columns of A

-every b in ℝ^m has a solution (Ax=b)

matrix equation

(A)(x)=(b)

vector equation

x₁a₁+x₂a₂+…+x_na_n=b

x=

A^-1b

A^-1=

(1/ad-bc) [ d -b ]

[ -c a ]

( A | I ) —>

( I | A^-1 )

in order to be linearly independent

the set msut have less vectors (columns) than entries in each vector (rows)

Vector Space Axiom One

for any u, v in V, u + v is also in V (sum of u + v is in V)

Vector Space Axiom Two

u + v = v + u (communative property of addition)

Vector Space Axiom Three

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

Vector Space Axiom Four

V has a vector 0 such that u + 0 = u (additive identity)

Vector Space Axiom Five

For each u in V, there is a vector -u in V such that u + (-u) = 0 (additive inverse)

Vector Space Axiom Six

For any scalar, c, the vector cu is in V

Vector Space Axiom Seven

c(u + v) = cu + cv

Vector Space Axiom Eight

(c + d)u = cu + du

Vector Space Axiom Nine

c(du) = (cd)u

Vector Space Axiom Ten

1u = u

Subspace Property One

the zero vector, 0, is in H

Subspace Property Two

whenever u and v are in H, u + v shoudl also be in H (addition)

Subspace Property Three

for any scalar, c, the vector cu is in H (scalar multiplication)

Null Space- Nul(A)

the set of all possible vectors that satisfy Ax = 0 fomring an entire vector space (total solution space)

subspace of lRⁿ

Column Space- Col(A)

the columns of the matrix Ex: { (), () }

subspace of lR^m

Coordinate Vector

[x]_B = c₁b₁ + c₂b₂ + … + c_nb_n

Basis for Column Space

the pivot columns of A

Basis for Row Space

the pivot rows of A

Nullity of A

number of non-pivot columns

dimension of Nul(A)

Rank of A

number of pivot columns

dimension of Row(A)/Col(A)

Dimension of Subspace

the number of vectors in the basis

Basis of Null Space

minimal set of linearly independent vectors that span the vector space (can span the space)

Determinant of Trigangular Matrix

the product of its diagonal entries

Determinant Property One: adding a multiple of a row to another

det(A) = det(B)

Determinant Property Two: interchange two rows

det(A) = -det(B)

Determinant Property Three: multiply any row by a number, k

det(A) = k det(A)

Determinant Property Four: |A| does equal 0

A^-1 exists

Determinant Property Five: |A^T|

= |A|

Determinant Property Six: |AB|

|A| |B|

Inverse Matrix Formula A^-1 =

¹/_det(A) adj(A)

adjugate formula

adj(A) = [C_ij]^T

also known as the transpose of the cofacotr matrix

Area of 2×2 matrix (Area of the parallelogram)

|det(A)|

Volume of 3×3 matrix (Volume of the parallelepiped)

|det(A)|

Nullity + Rank

= columns of A (meaning n)