# Study sets matching "linear algebra 2"

Study sets

Diagrams

Classes

Users

Vector Space

Vector Space Axioms

Subspace

Linear Combination

A nonempty set V of vectors following 10 axioms for vectors u,…

1: u + v is in V... 2: u + v = v + u... 3: (u + v) + w = u + (v + w)…

A subspace H of vector space V is a subset that has three prop…

Any sum of scalar multiples of vectors

Vector Space

A nonempty set V of vectors following 10 axioms for vectors u,…

Vector Space Axioms

1: u + v is in V... 2: u + v = v + u... 3: (u + v) + w = u + (v + w)…

orthogonal complement of a vector space

4 Fundamental Spaces

TA

domain

The set of all vectors orthogonal to all the vectors in a vect…

row(A) col(A) null(A) null(At)

matrix transformation

all possible inputs

orthogonal complement of a vector space

The set of all vectors orthogonal to all the vectors in a vect…

4 Fundamental Spaces

row(A) col(A) null(A) null(At)

Augmented Matrix

3x4 Matrix

Elementary Row Operations for Matrices

Row Equivalent Matrices

3 rows and 4 columns

(1) Interchange any two rows... (2) Multiply a row by any nonzero…

Matrices that have a sequence of row operations to transform o…

Augmented Matrix

3x4 Matrix

3 rows and 4 columns

associative property of addition

coefficient

commutative property of addition

constant

(a + b) + c = a + (b + c)

the number that is multiplied by the variable

a + b = b + a

a term that has no variable, a single number

associative property of addition

(a + b) + c = a + (b + c)

coefficient

the number that is multiplied by the variable

Scalar multiplication

Standard Basis in Rn

Closed under addition

Closed under scalar multiplication

Multiplication of vectors by t, such that it becomes t(v1), t(…

Denoted by

**e**n, [1, 0, 0, 0...0] to the nth number. So in R3,…Two vectors, when added together, are still in the same subspa…

When a vector is multiplied by a scalar, it's still in the sam…

Scalar multiplication

Multiplication of vectors by t, such that it becomes t(v1), t(…

Standard Basis in Rn

Denoted by

**e**n, [1, 0, 0, 0...0] to the nth number. So in R3,…Commutative Property of Addition

Commutative Property of Multiplication

Associative Property of Addition

Associative Property of Multiplication

-12+20+13=-12+13+20

d(ef)=d(fe)

6+(0+7)=(6+0)+7

(7∙5)6=7(5∙6)

Commutative Property of Addition

-12+20+13=-12+13+20

Commutative Property of Multiplication

d(ef)=d(fe)

Suppose A is nxn and the equation Ax=0…

Use the given inverse of the coefficien…

Test if matrix is invertible?

Find the LU factorization of the Matrix…

c. Suppose A is n x n and the equation Ax = 0 has only the tri…

two ways to solve ... 1. Ax=b --> x=[A]-1 b... 2. use equation mode

1. If determinant = 0. yes not =0 then no... 2. has pivot in ever…

1. solve for U first. (zeros on bottom)... 2. solve for L ... ......…

Suppose A is nxn and the equation Ax=0…

c. Suppose A is n x n and the equation Ax = 0 has only the tri…

Use the given inverse of the coefficien…

two ways to solve ... 1. Ax=b --> x=[A]-1 b... 2. use equation mode

an n-vector

equality of n-vectors

n-space, R^n

addition of n-vectors

A single column of numbers that has n entries (an n x 1 matrix)

two n-vectors are equal if and only if all their components ar…

the collection of all n-vectors

add the components of the vectors horizontally.

an n-vector

A single column of numbers that has n entries (an n x 1 matrix)

equality of n-vectors

two n-vectors are equal if and only if all their components ar…

Span/Spanning sequence

Criteria for spanning

Necessary condition for spanning R^n

Subset

Span: the set of all linear combinations Span(v₁, v₂,..., v_k)…

Let A be an n x k matrix with columns v₁, v₂,..., v_k:... 1. For…

Assume (v₁, v₂,..., v_k) is a spanning sequence of R^n then k…

Assume X and Y are sets... If all objects that in X also belongs…

Span/Spanning sequence

Span: the set of all linear combinations Span(v₁, v₂,..., v_k)…

Criteria for spanning

Let A be an n x k matrix with columns v₁, v₂,..., v_k:... 1. For…

Relation

Domain

Range

Function

A mapping or pairing of input values with output values.

The set of input values.

The set of output values.

A relation for which each input has exactly one output.

Relation

A mapping or pairing of input values with output values.

Domain

The set of input values.

Coordinate Plane

x-axis

y-axis

Origin

Formed by two perpendicular real number lines.

The horizontal line of the coordinate plane.

The vertical line of the coordinate plane.

The point at which the two lines meet.

Coordinate Plane

Formed by two perpendicular real number lines.

x-axis

The horizontal line of the coordinate plane.

Zero Matrix

-A in A+(-A)=0 is called the...

Diagonal Matrix

Identity Matrix

Matrix with all zeros

additive inverse of A

a square matrix whose non-diagonal entries are zeros

a square matrix with ones on the main diagonal and zeros every…

Zero Matrix

Matrix with all zeros

-A in A+(-A)=0 is called the...

additive inverse of A

Vector (Components)

n-space

Equivalent vectors

Sum/Add two n-vectors

Vector: single column of numbers... n-vector: n number of entrie…

Rⁿ: collection of all n-vectors

Two vectors are equal if and only if they have identical compo…

To sum or add two n-vectors u, v add the corresponding compone…

Vector (Components)

Vector: single column of numbers... n-vector: n number of entrie…

n-space

Rⁿ: collection of all n-vectors

Determinant Identity Matrix

Consequences of the Exchange Matrix Res…

Determinants When Multiplying by an Exc…

Corollary 4.2.5, 4.2.6

The determinant of the n x n identity matrix I_n is 1

Assume that 1 ≤ i < j ≤ n ... Then,... det(Pⁿ_(ij)) = -1 and... det(P_(…

Let A be an n x n matrix ... Assume that the matrix B is obtained…

If two rows of an n x n matrix A are equal:... det(A) = 0... If the…

Determinant Identity Matrix

The determinant of the n x n identity matrix I_n is 1

Consequences of the Exchange Matrix Res…

Assume that 1 ≤ i < j ≤ n ... Then,... det(Pⁿ_(ij)) = -1 and... det(P_(…

Matrix (Rows/Columns/Entry)

Zero row

Coefficient matrix

Augmented matrix

Matrix: rectangular array of numbers... Entry: numbers in the ma…

Zero row: all entries in matrix row = 0... Nonzero row: at least…

Matrix consisting of the coefficients of the variables in the…

Inhomogeneous linear system:... add column to matrix for constant…

Matrix (Rows/Columns/Entry)

Matrix: rectangular array of numbers... Entry: numbers in the ma…

Zero row

Zero row: all entries in matrix row = 0... Nonzero row: at least…

Dependence relation

Trivial/Nontrivial dependence relation

Linearly dependent/independent

Criteria for Linear independence

By a dependence relation on ... (v₁, v₂,..., v_k) we mean:... Any l…

Trivial dependence relation:... 0v₁ + 0v₂ +...+ 0v_k = 0_n... Non-t…

Linearly dependent: ... If there exists a non-trivial dependence…

1. (v₁, v₂,..., v_k) is linearly independent... 2. homogeneous l…

Dependence relation

By a dependence relation on ... (v₁, v₂,..., v_k) we mean:... Any l…

Trivial/Nontrivial dependence relation

Trivial dependence relation:... 0v₁ + 0v₂ +...+ 0v_k = 0_n... Non-t…

Kernel of a Linear Transformation

Kernel of a Linear Transformation Theor…

Rank of Linear Transformation

Nullity of Linear Transformation

Let T : V → W be a linear transformation... Kernel of T: Consist…

Let T : V → W be a linear transformation... Then, Ker(T) is a su…

Let V and W be finite dimensional vector spaces and T : V → W…

Let V and W be finite dimensional vector spaces and T : V → W…

Kernel of a Linear Transformation

Let T : V → W be a linear transformation... Kernel of T: Consist…

Kernel of a Linear Transformation Theor…

Let T : V → W be a linear transformation... Then, Ker(T) is a su…

Span of a Vector Space

Spanning Sequence/Set of Vector Space

Span of Sequence is a Subspace

Union of Two Sequences

Let V be a vector space and (v₁, v₂,..., v_k) be a sequence of…

Let V be a vector space and W a subspace of V: ... Spanning Sequ…

Let S be either a sequence or a set from a vector space V.... The…

Let A = (u₁, u₂, u_k) and B = (v₁, v₂, v_l) be two sequences o…

Span of a Vector Space

Let V be a vector space and (v₁, v₂,..., v_k) be a sequence of…

Spanning Sequence/Set of Vector Space

Let V be a vector space and W a subspace of V: ... Spanning Sequ…

Dot product

Perpendicular/Orthogonal

Properties of the dot product

Theorem 2.6.2

Let u = (u₁ ) v = (v₁ ) be 2 n-vectors... (u₂ ) (v₂ ) ... (... ) (…

Perpendicular/Orthogonal:... When the dor product of two vectors…

Let u, v, w be n-vectors and c any scalar... 1. Symmetry: u.v =…

Let v₁, v₂,..., v_m, u be n-vectors and ... c₁, c₂,... c_m be sca…

Dot product

Let u = (u₁ ) v = (v₁ ) be 2 n-vectors... (u₂ ) (v₂ ) ... (... ) (…

Perpendicular/Orthogonal

Perpendicular/Orthogonal:... When the dor product of two vectors…

Zero/Nonzero Subspace

Another way to construct a subspace

Basis

Existence basis theorem

Zero subspace:... The subspace {0_n} of R^n... Nonzero subspace:... An…

Let (v₁,..., v_m) be a sequence of n-vectors... Let V consist of…

Let V be a nonzero subspace of R^n.... A sequence β of V is a bas…

Let W be a nonzero subspace of R^n.... Then W has a basis... (with a…

Zero/Nonzero Subspace

Zero subspace:... The subspace {0_n} of R^n... Nonzero subspace:... An…

Another way to construct a subspace

Let (v₁,..., v_m) be a sequence of n-vectors... Let V consist of…