Open Educational Resources

Linear Vector Spaces: Additional Structure for Linear Vector Spaces

Learning Outcomes

  • Calculate the norm of vectors, the dot product, and the distance between vectors
  • Identify the relationship between the Mathematical definition and the associated geometry of linear vector spaces

Norm

Vector spaces can be endowed with a function called the “Norm” function. The norm function is used to describe the size of vectors in a linear vector space. It assigns strictly positive values for nonzero vectors.

Let V be a linear vector space over \mathbb{R}. A norm is defined as a function \|\text{ }\|:V \rightarrow [0,\infty) that satisfies the following properties. \forall v, u \in V,\forall \alpha \in \mathbb{R}:

  1. Positive homogeneity or positive scalability: \| \alpha v\|=|\alpha|\|v\|
  2. Triangle inequality: \| u+v\|\leq \|u\|+\|v\|
  3. Positive definiteness: \|u\|=0\Leftrightarrow u=0

There are many candidate functions that satisfy the above definition. The “Euclidean Norm” is by far the most commonly used norm function. The Euclidean norm is defined as:

    \[\forall v\in \mathbb{R}^n,\|v\|=\sqrt{v_1^2+v_2^2+\cdots+v_n^2}\]

Exercises:
  • Show that the triangle inequality is equivalent to: \left|{\|u\|-\|v\|}\right|\leq \|u-v\|
  • Show that the Euclidean norm satisfies the properties of a norm function
  • Show that the function |||\text{ }|||:\mathbb{R}^n \rightarrow [0,\infty) such that \forall v\in \mathbb{R}^n,|||v|||=\max\{v_1,v_2,\cdots,v_n\} satisfies the properties of a norm function

Metric (Distance)

Vector spaces can be endowed with a “Metric” function. The metric function is used to define a distance between any two vectors in a linear vector space. Let V be a linear vector space over \mathbb{R}. A metric is defined as a function \rho:V\times V \rightarrow [0,\infty) that satisfies the following properties. \forall x, y, z \in V:

  1. Symmetry: \rho(x,y)=\rho(y,x)
  2. Triangle inequality: \rho(x,z)\leq \rho(x,y)+\rho(y,z)
  3. \rho(x,y)=0 \Leftrightarrow x=y

There are many candidate functions that satisfy the above definition. Usually, the metric function can be induced from the norm function by setting \rho(x,y)=\|x-y\|. The “Euclidean Metric” is a commonly used metric function and is generated by the “Euclidean Norm”:

    \[ \forall x,y\in \mathbb{R}^n,\rho(x,y)=\|x-y\|=\sqrt{(x_1 - y_1)^2+(x_2-y_2)^2+\cdots+(x_n-y_n)^2} \]

Exercises:
  • Show that the Euclidean metric satisfies the properties of a metric function.
  • Show that the function \rho:V\times V \rightarrow [0,\infty) defined as: \rho(x,y)=\|x-y\| satisfies the properties of a metric function.

Inner Product

Vector spaces can be endowed with a function called the Inner Product function which associates each pair of vectors with a scalar quantity. The inner product function allows adding geometric structure to the vector space by defining orthogonality and angles. Let V be an linear vector spaces over the field of complex numbers \mathbb{C}. The inner product is a map \langle \text{ , } \rangle:V\times V\rightarrow \mathbb{C} such that \forall x,y,z\in V, \forall \alpha,\beta \in \mathbb{C}:

  1. Conjugate symmetry: \langle x,y \rangle=\overline{\langle y,x \rangle}
  2. Linearity in the first argument: \langle \alpha x+\beta y,z \rangle=\alpha\langle x,z \rangle +\beta\langle y,z \rangle
  3. Positive definiteness: \langle x,x \rangle \geq 0 with \langle x,x \rangle = 0\Leftrightarrow x=0

When V is defined over the field of real numbers \mathbb{R}, the inner product becomes a bilinear map and conjugate symmetry becomes symmetry.

Euclidian Dot Product

The “Euclidean Dot Product” is one example of an inner product function. The dot product is defined \forall x,y\in\mathbb{R}^n as follows:

    \[ \langle x,y \rangle =x\cdot y=x_1 y_1+x_2 y_2+\cdots+x_n y_n \]

The “Euclidean Dot Product” is related to the “Euclidean Norm” as follows, \forall x \in \mathbb{R}^n:

    \[ \|x\|^2=x\cdot x \]

Euclidian Dot Product in \mathbb{R}^2 and \mathbb{R}^3

In two and three dimensional vector spaces, the dot product is related to the geometric angle between two vectors as follows, \forall x,y\in\mathbb{R}^3

    \[ x\cdot y = x_1 y_1+x_2 y_2+x_3 y_3=\|x\|\|y\|\cos\theta \]

where \theta is the geometric angle between the vectors x and y.

Orthogonal Vectors

Two vectors are called “orthogonal” if their inner product is equal to zero. For Euclidean vector spaces, the term: “perpendicular” is used interchangeably with “orthogonal”.

Exercises:
  • Show that {x_1} {y_1} +x_2 y_2+x_3 y_3=\|x\|\|y\|\cos\theta

    Video:

    Quiz 4- Additional Structure:

    Solution guide

    Leave a Reply

    Your email address will not be published.