Introduction to Vector Spaces

Vector spaces consist of set $E$ with vector addition and scalar multiplication operations.
Vector addition is defined as +: $E \times E \rightarrow E$, where $(u, v) \mapsto u + v$.
Scalar multiplication is defined as ⋅: $\mathbb{K} \times E \rightarrow E$, where $(\lambda, u) \mapsto \lambda \cdot u = \lambda u$ and $\mathbb{K}$ is a field (usually $\mathbb{R}$ or $\mathbb{C}$).
Associativity of addition: $(u + v) + w = u + (v + w)$.
Commutativity of addition: $u + v = v + u$.
Additive identity: There exists $0 \in E$ such that $u + 0 = u$.
Additive inverse: There exists $-u \in E$ such that $u + (-u) = 0$.
Compatibility of scalar multiplication: $\lambda \cdot (\mu \cdot u) = (\lambda \mu) \cdot u$.
Multiplicative identity: $1 \cdot u = u$.
Distributivity of scalar multiplication over vector addition: $\lambda \cdot (u + v) = \lambda \cdot u + \lambda \cdot v$.
Distributivity of scalar multiplication over field addition: $(\lambda + \mu) \cdot u = \lambda \cdot u + \mu \cdot u$.
$\mathbb{R}^n$ is a vector space: n-tuples of real numbers with standard addition and scalar multiplication.
$\mathbb{C}^n$ is a vector space: n-tuples of complex numbers with standard addition and scalar multiplication.
$\mathcal{M}_{m,n}(\mathbb{K})$ is a vector space: $m \times n$ matrices with entries in $\mathbb{K}$ with standard matrix addition and scalar multiplication.
$\mathbb{K}[X]$ is a vector space: polynomials with coefficients in $\mathbb{K}$ with standard polynomial addition and scalar multiplication.
$\mathcal{F}(E, F)$ is a vector space: functions from $E$ to $F$ (where $F$ is a vector space over $\mathbb{K}$) with standard function addition and scalar multiplication.
A subset $F$ of a vector space $E$ is a subspace if it is non-empty, closed under addition, and closed under scalar multiplication.
Subspaces are stable under linear combinations
Lines through the origin in $\mathbb{R}^2$ are subspaces.
Planes through the origin in $\mathbb{R}^3$ are subspaces.
Polynomials of degree less than or equal to $n$ form a subspace of $\mathbb{K}[X]$.
Continuous functions on $[a, b]$ form a subspace of all functions on $[a, b]$.

Linear Combinations

A linear combination of vectors $v_1, v_2,..., v_n \in E$ is a vector of the form $\lambda_1 v_1 + \lambda_2 v_2 +... + \lambda_n v_n$ where $\lambda_1, \lambda_2,..., \lambda_n \in \mathbb{K}$.

Span

The span (or linear hull) of a set $S \subseteq E$, denoted $Vect(S)$, is the set of all possible linear combinations of vectors in $S$: $Vect(S) = { \lambda_1 v_1 + \lambda_2 v_2 +... + \lambda_n v_n \mid v_1, v_2,..., v_n \in S, \lambda_1, \lambda_2,..., \lambda_n \in \mathbb{K}, n \in \mathbb{N} }$.
$Vect(S)$ is the smallest subspace of $E$ containing $S$.

Linear Independence

The set of vectors $(v_1, v_2,..., v_n)$ is linearly independent if the only linear combination that equals zero is when all coefficients are zero: $\lambda_1 v_1 + \lambda_2 v_2 +... + \lambda_n v_n = 0 \Rightarrow \lambda_1 = \lambda_2 =... = \lambda_n = 0$.

Spanning Set

The set of vectors $(v_1, v_2,..., v_n)$ is a spanning set if every vector in $E$ can be written as a linear combination of these vectors: For all $v \in E$, there exist $\lambda_1, \lambda_2,..., \lambda_n \in \mathbb{K}$ such that $v = \lambda_1 v_1 + \lambda_2 v_2 +... + \lambda_n v_n$.

Basis

A basis of $E$ is a set of vectors that is both linearly independent and a spanning set.

Dimension

If $E$ has a finite basis, then all bases of $E$ have the same number of elements called the dimension of $E$, denoted $dim(E)$.
$dim(\mathbb{R}^n) = n$.
$dim(\mathbb{C}^n) = n$.
$dim(\mathcal{M}_{m,n}(\mathbb{K})) = mn$.
$dim(\mathbb{K}_n[X]) = n + 1$, where $\mathbb{K}_n[X]$ is the set of polynomials of degree less than or equal to $n$.
If $F$ is a subspace of $E$, then $dim(F) \leq dim(E)$.
If $dim(F) = dim(E) < \infty$, then $F = E$.

- The androecium comprises stamens, the male reproductive structures in a flower. - Each stamen has a stalk (filament) and an anther. - An anther typically has two lobes, each containing two pollen sacs where pollen grains are produced. - A sterile stamen is called a staminode. - Stamens may unite with other floral parts, such as petals, or among themselves. - Stamens attached to petals are epipetalous (e.g., in brinjal) - Stamens attached to the perianth are epiphyllous (e.g., lily). - Stamens may be free (polyandrous) or united to varying degrees. - Stamens may unite into one bunch or bundle, known as monoadelphous (e.g., China rose) - Stamens may unite into two bundles, known as diadelphous (e.g., pea) - Stamens may unite into more than two bundles, known as polyadelphous (e.g., citrus). - Filament length may vary within a flower (e.g., Salvia, mustard). - The gynoecium constitutes the female reproductive part of a flower, composed of one or more carpels. - A carpel consists of the stigma, style, and ovary. - The ovary is the enlarged basal part where the elongated style lies. - The style connects the ovary to the stigma. - The stigma, usually at the tip of the style, serves as the receptive surface for pollen grains. - The ovary contains one or more ovules attached to a placenta. - Multiple free carpels are termed apocarpous (e.g., lotus, rose). - Fused carpels are termed syncarpous (e.g., mustard, tomato). - After fertilization, ovules become seeds, and the ovary matures into a fruit. - Placentation refers to the arrangement of ovules within the ovary. - Placentation types: marginal, axile, parietal, basal, central, and free central. - Marginal placentation: the placenta forms a ridge along the ventral suture of the ovary, bearing ovules in two rows (e.g., pea). - Axile placentation: the placenta is axial with ovules attached in a multilocular ovary (e.g., China rose, tomato, lemon). - Parietal placentation: ovules develop on the inner wall or peripheral part of the ovary, which is one-chambered but may become two-chambered due to false septum formation (e.g., mustard, Argemone). - Septa are Absent: ovules are borne on a central axis (e.g., Dianthus, Primrose).