Bio 5.8 easy

Linear Algebra

Definition 1.1 (Vector Space)

• A real vector space E is a set with two operations:
  • Addition: (E \times E \rightarrow E), ((u, v) \mapsto u + v)
  • Scalar multiplication: (\mathbb{R} \times E \rightarrow E), ((\lambda, u) \mapsto \lambda u)
• These operations must satisfy the following properties:
  • Associativity of addition: (\forall u, v, w \in E), ((u + v) + w = u + (v + w))
  • Commutativity of addition: (\forall u, v \in E), (u + v = v + u)
  • Existence of an additive identity: (\exists 0_E \in E, \forall u \in E, u + 0_E = u)
  • Existence of an additive inverse: (\forall u \in E, \exists v \in E, u + v = 0_E). (v = -u)
  • Distributivity of scalar multiplication over vector addition: (\forall \lambda \in \mathbb{R}, \forall u, v \in E), (\lambda(u + v) = \lambda u + \lambda v)
  • Distributivity of scalar multiplication over scalar addition: (\forall \lambda, \mu \in \mathbb{R}, \forall u \in E), ((\lambda + \mu)u = \lambda u + \mu u)
  • Compatibility of scalar multiplication with real multiplication: (\forall \lambda, \mu \in \mathbb{R}, \forall u \in E), (\lambda(\mu u) = (\lambda \mu)u)
  • Identity element for scalar multiplication: (\forall u \in E, 1u = u)

Definition 1.2 (Vector Subspace)

• Given a vector space (E), a subset (F \subseteq E) is a subspace if:
  • (F) is non-empty.
  • (\forall u, v \in F), (u + v \in F)
  • (\forall \lambda \in \mathbb{R}, \forall u \in F), (\lambda u \in F)

Definition 1.3 (Linear Combination)

• Given vectors (u_1, ..., u_n \in E), a linear combination is a vector of the form:
  • (\lambda_1 u_1 + ... + \lambda_n u_n), where (\lambda_1, ..., \lambda_n \in \mathbb{R})

Definition 1.4 (Spanned Vector Space)

• Given vectors (u_1, ..., u_n \in E), the space spanned by them, (Vect(u_1, ..., u_n)), is the set of all their linear combinations.
  • (Vect(u_1, ..., u_n) = {\lambda_1 u_1 + ... + \lambda_n u_n \mid \lambda_1, ..., \lambda_n \in \mathbb{R}})

Definition 1.5 (Free Family)

• A family of vectors ((u_1, ..., u_n)) is free (linearly independent) if:
  • (\lambda_1 u_1 + ... + \lambda_n u_n = 0_E \Rightarrow \lambda_1 = ... = \lambda_n = 0)

Definition 1.6 (Generating Family)

• A family of vectors ((u_1, ..., u_n)) is generating if any vector in (E) can be written as a linear combination of them.
  • (\forall u \in E, \exists \lambda_1, ..., \lambda_n \in \mathbb{R}, u = \lambda_1 u_1 + ... + \lambda_n u_n)

Definition 1.7 (Basis)

• A basis of (E) is a family of vectors that is both free and generating.

Definition 1.8 (Dimension)

• If (E) has a finite basis, all bases of (E) have the same number of elements. This number is the dimension of (E), denoted (dim(E)).

Definition 1.9 (Linear Transformation)

• Given vector spaces (E) and (F), a map (f: E \rightarrow F) is linear if:
  • (\forall u, v \in E), (f(u + v) = f(u) + f(v))
  • (\forall \lambda \in \mathbb{R}, \forall u \in E), (f(\lambda u) = \lambda f(u))

Definition 1.10 (Kernel)

• The kernel of (f), (Ker(f)), is the set of vectors in (E) mapped to (0_F) by (f).
  • (Ker(f) = {u \in E \mid f(u) = 0_F})

Definition 1.11 (Image)

• The image of (f), (Im(f)), is the set of vectors in (F) reached by (f).
  • (Im(f) = {v \in F \mid \exists u \in E, f(u) = v})

Properties

Proposition 2.1

• (F) is a subspace of (E) if and only if:
  • (0_E \in F)
  • (\forall u, v \in F), (u + v \in F)
  • (\forall \lambda \in \mathbb{R}, \forall u \in F), (\lambda u \in F)

Proposition 2.2

• (Ker(f)) is a subspace of (E) and (Im(f)) is a subspace of (F).

Theorem 2.3 (Rank Theorem)

• Given a linear transformation (f: E \rightarrow F), if (E) is finite-dimensional, then:
  • (dim(E) = dim(Ker(f)) + dim(Im(f)))

Proposition 2.4

• Given a linear transformation (f: E \rightarrow F):
  • (f) is injective (\Leftrightarrow Ker(f) = {0_E})
  • (f) is surjective (\Leftrightarrow Im(f) = F)
  • (f) is bijective (\Leftrightarrow Ker(f) = {0_E}) and (Im(f) = F)

Division Rules

• For dividing two natural numbers (D and d), find a third natural number (c) that, when multiplied by d, equals D: (D = d \cdot c)
  • D = Dividend
  • d = Divisor
  • c = Quotient
• Example: (15 : 5 = 3) because (15 = 5 \cdot 3)

Division with Integers

• To divide two integers, divide their absolute values.
• The sign of the result is "+" if the signs of the two numbers being divided are the same, and "-" if they are different.
  • ((+15) : (+5) = +3)
  • ((-15) : (-5) = +3)
  • ((+15) : (-5) = -3)
  • ((-15) : (+5) = -3)

Priority of Operations

• When multiple operations are in a sequence, follow this order:
  • Parentheses
  • Exponents and roots
  • Multiplications and divisions
  • Additions and subtractions
• If there are multiple operations of the same priority in a row, perform them from left to right.
• Example: (10 - 2 \cdot (15 : 5) + 3 = 10 - 2 \cdot 3 + 3 = 10 - 6 + 3 = 4 + 3 = 7)

Other Important Points

• Division by zero is not possible.
• Dividing by 1 results in the same number.