1. Introduction Vectors are fundamental objects in mathematics and physics. They represent quantities that have both magnitude and direction, such as displacement, velocity, and force. In this section, we will explore the properties of vectors and their applications in various fields.
1.1. Types of Vectors There are several types of vectors, including:
- Displacement vectors, which represent the change in position of an object.
- Velocity vectors, which represent the rate of change of position of an object.
- Force vectors, which represent the interaction between two objects.
1.2. Vector Operations Vectors can be added together to form a new vector, and they can be multiplied by a scalar to produce a new vector. These operations are defined as follows:
- Vector Addition: If $\mathbf{u}$ and $\mathbf{v}$ are vectors, then their sum $\mathbf{u} + \mathbf{v}$ is a vector that represents the combined effect of $\mathbf{u}$ and $\mathbf{v}$.
- Scalar Multiplication: If $\mathbf{u}$ is a vector and $a$ is a scalar, then the product $a\mathbf{u}$ is a vector that represents the scaling of $\mathbf{u}$ by a factor of $a$.
1.3. Vector Spaces A vector space is a set of vectors that is closed under addition and scalar multiplication. Formally, a vector space $V$ over a field $F$ is a set of elements of $V$ that is closed under addition and scalar multiplication. In other words, if $\mathbf{u}$ and $\mathbf{v}$ are vectors in $V$ and $a$ is a scalar in $F$, then $\mathbf{u}+\mathbf{v}$ and $a\mathbf{u}$ are also vectors in $V$.
1.4. Linear Independence A set of vectors ${\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n}$ is said to be linearly independent if the only way to write the zero vector as a linear combination of the $\mathbf{v}_i$’s is to set all the coefficients to zero. In other words, if $a_1\mathbf{v}_1 + a_2\mathbf{v}_2 + \ldots + a_n\mathbf{v}_n = \mathbf{0}$, then $a_1 = a_2 = \ldots = a_n = 0$.
1.5. Basis and Dimension A basis for a vector space $V$ is a set of linearly independent vectors that span $V$. The dimension of $V$ is the number of vectors in a basis for $V$. For example, the standard basis for $\mathbb{R}^n$ is the set of vectors ${\mathbf{e}_1, \mathbf{e}_2, \ldots, \mathbf{e}_n}$, where $\mathbf{e}_i$ has a $1$ in the $i$-th position and $0$’s elsewhere.
In the context of vectors, the concept of measurability is not typically used as in the context of measure theory. However, if we were to adapt the concept, we might define it as follows:
Using this adapted concept of measurability, we could discuss the measurability of vectors in a measurable space. For example, if we have a measurable space $(X, \mathcal{M})$ and a vector space $V$ over a field $F$, we could consider a map $f: X \to V$ and discuss whether $f$ is measurable according to the above definition.
