Denis Gorodkov

Vector space (over real numbers)

4 min read

\underset{\substack{\uparrow\ \text{vector}}}{v}

You can't use 'macro parameter character #' in math mode> The same applies to $0$: be careful to determine in which case it is the $0$ vector or the real number $0$. Sometimes the $0$ vector is written in bold or with an arrow on top following the convention for vector notation. > [!d] Definition: *abstract vector space* > > A *(real) vector space* is a non-empty set $V$ of objects called *vectors* with the following operations: > 1. **Vector addition operation ($+$):** A binary operation that combines two vectors in the set $V$, producing another vector in $V$. > 2. **Multiplication by scalar operation ($\cdot$ or omitted):** A binary operation that multiplies a vector by a real number (usually called *scalars* in this setting), yielding another vector in $V$. >The vector space must satisfy the following properties, known as vector space *axioms*: >- **Closure under addition:** The sum of any two vectors in $V$ is also in $V$. >- **Commutativity of addition:** $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$ for all $\mathbf{u}, \mathbf{v}$ in $V$. >- **Associativity of addition:** $(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$ for all $\mathbf{u}, \mathbf{v}, \mathbf{w}$ in $V$. >- **Existence of additive identity:** There exists a zero vector $\mathbf{0}$ in $V$ such that $\mathbf{v} + \mathbf{0} = \mathbf{v}$ for all $\mathbf{v}$ in $V$. >- **Existence of additive inverse:** For every vector $\mathbf{v}$ in $V$, there exists a vector $\mathbf{-v}$ in $V$ such that $\mathbf{v} + \mathbf{(-v)} = \mathbf{0}$. >- **Closure under multiplication by scalar:** The product of any scalar and any vector in $V$ is also in $V$. >- **Compatibility of multiplication by scalar with real number multiplication:** $a \cdot (b \cdot \mathbf{v}) = (a \cdot b) \cdot \mathbf{v}$ for all scalars $a, b$ and all vectors $\mathbf{v}$ in $V$. >- **Identity element of multiplication by scalar:** For all $\mathbf{v}$ in $V$ it is true that $1 \cdot \mathbf{v} = \mathbf{v}$. >- **Distributivity of multiplication by scalar over vector addition:** $a \cdot (\mathbf{u} + \mathbf{v}) = a \cdot \mathbf{u} + a \cdot \mathbf{v}$ for all scalars $a$ and all vectors $\mathbf{u}, \mathbf{v}$ in $V$. >- **Distributivity of multiplication by scalar over real number sum:** $(a + b) \cdot \mathbf{v} = a \cdot \mathbf{v} + b \cdot \mathbf{v}$ for all scalars $a, b$ and all vectors $\mathbf{v}$ in $V$. ^c30824 This definition, contrarily to the lecture one, is specifically worded without using quantifiers for clarity (here is a [[Vector-space-(over-real-numbers)-quantifier-definition|link]] to the definition using quantifiers). It is a good exercise in learning quantifiers to try to recall how to write these in quantifiers seeing the text version and vice versa. ## Why is this definition even introduced? 1. It generalizes several examples: vectors in the plane $\mathbb{R}^2$ and the space $\mathbb{R}^3$, [[Teaching/MAT-224/Lecture-notes/Bits/Polynomials|polynomials with real coefficients]], functions with real values. 2. It allows to prove complicated (but practically useful!) results for all these examples at once, and these proofs are easier as the concrete examples are stripped of the unnecessary details. 3. It unveils the similarity between the examples, and allows for a structural understanding of similar behavior and theorems in the individual examples. ## Examples - The Euclidean space $\mathbb{R}^n$, where vectors are $n$-tuples of real numbers. - The set of all polynomials of degree at most $n$ with real coefficients. - The set of all real-valued functions with a common domain. - The set of all real-valued sequences. - The set of all finite ($0$ from some point) real-valued sequences.

Graph View

Backlinks