Linear Algebra Exam 1 Definitions & Theorems

Linear Equation

An algebraic equation in which the variables that appear are only to degree one.

A system ℒ

A system ℒ ("script L") of linear equations is a set of linear equations that are supposed to hold simultaneously.

Solution to ℒ

A solution to ℒ is an assignment of variables making all the equations true simultaneously.

S(ℒ)

If ℒ is a system of linear equations, S(ℒ) is the set of all solutions to ℒ.

ℒ is consistent

ℒ is consistent if and only if it has a solution, i.e. S(ℒ) ≠ ∅ (empty set).

ℒ is inconsistent

ℒ is inconsistent if and only if S(ℒ) = ∅ (empty set).

ℒ₁ and ℒ₂ are equivalent

ℒ₁ and ℒ₂ are equivalent if and only if S(ℒ₁) = S(ℒ₂).

Matrix

A rectangular array of numbers.

Elementary Row Operations

1) Add a multiple of a row to another row
2) Swap 2 rows
3) Multiply a row through by a nonzero number (scaling)

Row Equivalent

Two matrixes of the same size are called row Equivalent if you can get from one to the other by doing elementary row operations.

Theorem: If the augmented matrices of ℒ₁ and ℒ₂ are row equivalent, then…

They have the same solution sets: S(ℒ₁) = S(ℒ₂).

Zero row, zero column, leading term

• A row that is all zeroes

• A column that is all zeroes

• The leading term of a nonzero row is the left most nonzero term in the row

REF

A matrix M is in REF iff:

1) all zero rows (if any) are at the bottom

2) any leading term of a row is to the right of all leading terms in the rows above it

3) all entries in the column below a leading term are zero

RREF

M is in RREF iff M is in REF and:

4) all leading terms equal 1

5) all entries in the column above a leading term are zero

Unique Matrix Theorem

Given M, it is row equivalent to a unique matrix in RREF. REFs can be different; the RREF is always unique.

The number of pivots, pivot positions, pivot rows, and pivot columns are all uniquely determined by the original matrix M.

Poison Row

A poison row is all zeros except a nonzero entry in the last positino

The Trichotomy

Given a system of linear equations ℒ, one of the three must occur:

1) ℒ has no solutions → inconsistent

2) ℒ has exactly one solution → consistent

3) ℒ has infinitely many solutions → consistent

ℒ is inconsistent (poison row version)

ℒ is inconsistent iff its augmented matrix has an REF with one or more poison rows.

A vector

A matrix with one column

Linear Combination

Suppose v₁, …, vₚ ∈ ℝⁿ and c₁, …, cₚ ∈ ℝ. Then c₁v₁ + … + cₚvₚ ∈ ℝⁿ is called "the linear combination of v₁, …, vₚ with weights c₁, …, cₚ."

b ∈ Span(v₁, …, vₚ) Theorem

Let v₁, …, vₚ, b ∈ ℝⁿ. Then b ∈ Span(v₁, …, vₚ) iff [v₁ … vₚ | b] is the augmented matrix of a consistent system, i.e. the REF of that matrix has no poison rows.

m x n matrix

Let A be a matrix w/ m rows and n columns. Say A is an m x n matrix

Identity Matrix

Iₙ = the n × n square matrix with everything 0 except 1s down the diagonal.

Ax = b

b ∈ Span(columns of A) ⟺ b is a linear combination of the columns of A ⟺ Ax = b has a solution ⟺ the REF of [A | b] has no poison rows.

Let A be an m × n matrix Theorem

The following are equivalent:

1) For all b ∈ ℝᵐ, Ax = b has a solution

2) For all b ∈ ℝᵐ, b is a linear combination of the columns of A

3) The columns of A span ℝᵐ

4) The REF of A has a pivot in every row (no zero rows)

Algebraic Properties of Ac

1) A(c + d) = Ac + Ad

2) A(λc) = λ(Ac)

For ℒ: Ax = b, Homogeneous Case

b = 0

Always consistent because there is never a poison row.

For Ax = 0, x = 0 is always a solution ("the trivial solution").

Inhomogeneous Case

Ax = b, b ≠ 0

There may be no solutions (if there is a poison row).

Suppose there are solutions, say Ap = b; then p is "a particular solution."

Nontrivial Solution Theorum

Ax = 0 has a nontrivial solution ⟺ it has infinitely many solutions (either only the trivial one or infinitely many) ⟺ it has one or more free variables (i.e. the REF of A has at least one column without a pivot).

If ℒ is homogeneous, S(ℒ) is the span of a set of "basic" vectors.

Nullspace

Nul(A) = S(Ax = 0) = {v | Av = 0}

Nullspace + Thoerum

If p is a particular solution (Ap = b), then S(Ax = b) = p + Nul(A) = {p + w | w ∈ Nul(A)}.

0 is always in the null space.

Linear Dependence

v₁, …, vₚ are linearly dependent iff c₁v₁ + … + cₚvₚ = 0 has a nontrivial solution ⟺ it has free variables.

Linearly Independent

v₁, …, vₚ are linearly independent iff the only solution to c₁v₁ + … + cₚvₚ = 0 is the trivial solution (c₁ = … = cₚ = 0).

Shortcuts:

1) 0 by itself is linearly dependent because 1·0 = 0

2) 0, v₂, v₃, v₄ is linearly dependent

3) If some vectors are linearly dependent and we add more vectors to them, the new set stays linearly dependent

4) If a set of vectors is linearly independent, then any subset is also linearly independent

5) A single vector v is linearly dependent iff v = 0

6) v₁, v₂ are linearly dependent ⟺ one of them is a scalar multiple of the other

Linearly Dependent Theorems 1 and 2

1) Let v₁, …, vₚ ∈ ℝᵐ. They are linearly dependent ⟺ one of them is a linear combination of the others.

2) Let v₁, …, vₚ ∈ ℝᵐ. If p > m, they are automatically linearly dependent (p = # of vectors > m = max # of pivots).

Domain and Codomain

Let f: S → U be a map.

Domain(f) = S

Codomain(f) = U

The domain is the inputs. The codomain is where the outputs land.

Image Definition

If s ∈ S, f(s) is the "image" of s under f.

Pre-image and Range Definition

If u ∈ U, the pre-image of u = {s ∈ S | f(s) = u}.

Range(f) = {f(s) | s ∈ S}, "everything that gets outputted."

Onto

f is onto iff Range(f) = U.

One-to-one definition

f is one-to-one iff f(s₁) = f(s₂) ⟹ s₁ = s₂.

Kernel

The kernel of T = Ker(T) = {x ∈ ℝⁿ | T(x) = 0} = the preimage of 0.

One-to-one Kernel Theorem

If T: ℝⁿ → ℝᵐ is linear, T is one-to-one iff Ker(T) = {0}.

Standard Basic Vectors

The standard basis vectors e₁, …, eₙ of ℝⁿ are the columns of Iₙ = the n × n identity matrix.

Standard Matrix of T Theorem

Let T: ℝⁿ → ℝᵐ be a linear map. Then there exists a unique m × n matrix A such that T(x) = Ax for all x ∈ ℝⁿ. In fact A = [T(e₁) … T(eₙ)].

A = "the standard matrix of T."

Summary Theorem

Let T: ℝⁿ → ℝᵐ be linear, with standard matrix A (A is an m × n matrix and T(x) = Ax).

1) T is onto iff for all b ∈ ℝᵐ, T(x) = b has a solution

2) T is onto iff the columns of A span ℝᵐ, which holds iff REF(A) has a pivot in every row

3) T is one-to-one ⟺ (T(x) = T(y) ⟹ x = y) ⟺ Ker(T) = {0}, where Ker(T) = {x ∈ ℝⁿ | T(x) = 0}

4) T is one-to-one iff Nul(A) = {0}

Matrix Multiplication

A is m × p, B is p × n. Write B = [b₁ … bₙ].

Then AB = [Ab₁ Ab₂ … Abₙ].

Standard Matrix of UT

Suppose A is the standard matrix of T, so T(x) = Ax.

Suppose B is the standard matrix of U, so U(y) = By.

(UT)(x) = U(T(x)) = U(Ax) = B(Ax) = (BA)x

So BA is the standard matrix of UT.

Transpose

Let A be an m × n matrix. Aᵀ = A transpose = the n × m matrix you get from A by swapping rows and columns.

Transpose Theorums

a) (Aᵀ)ᵀ = A

b) (A + B)ᵀ = Aᵀ + Bᵀ

c) (λA)ᵀ = λAᵀ

d) (AB)ᵀ = BᵀAᵀ