Defining Length in Vector Spaces: The Norm and Its Fundamental Properties

1. The Definition of Norm (Length)

In standard Euclidean geometry (like high school math), we find the length of a vector using the Pythagorean theorem. In abstract linear algebra, we generalize this concept using the Norm.

The Definition:

Let \(V\) be an inner product space. For any vector \(x \in V\), we define the norm (or length) of \(x\) as:

$$
||x|| = \sqrt{\langle x, x \rangle}
$$

A Simple Example:

Consider the vector space \(\mathbb{R}^2\) (the standard 2D plane) with the standard dot product.

Let \(x = (3, 4)\).

$$
\langle x, x \rangle = 3^2 + 4^2 = 9 + 16 = 25
$$

$$
||x|| = \sqrt{25} = 5
$$

So, the norm (length) of the vector \((3, 4)\) is \(5\).

2. Fundamental Properties of the Norm (Theorem 6.2)

Just defining a norm isn’t enough; we need to ensure it behaves the way “length” should behave intuitively. Theorem 6.2 guarantees this.

The Theorem:

Let \(V\) be an inner product space over a field \(F\). For all vectors \(x, y \in V\) and scalar \(c \in F\):

1. \(||cx|| = |c| \cdot ||x||\)
2. \(||x|| = 0\) if and only if \(x = 0\). (And \(||x|| \ge 0\) always).
3. Cauchy-Schwarz Inequality: \(||\langle x, y \rangle|| \le ||x|| \cdot ||y||\)
4. Triangle Inequality: \(||x + y|| \le ||x|| + ||y||\)

The Meaning:

• Property (a) – Scaling: If you stretch a vector by a factor of \(c\) (e.g., doubling it), its length scales by the absolute value of \(c\). If you double a vector, its length doubles.
• Property (b) – Positivity: Lengths are never negative. The only object with zero size is the zero vector itself.
• Property (c) – Cauchy-Schwarz: This is arguably the most famous inequality in analysis. It states that the inner product of two vectors can never exceed the product of their lengths. This essentially puts a “bound” on the inner product, which allows us to define angles between vectors (since the cosine of an angle must be between -1 and 1).
• Property (d) – Triangle Inequality: The shortest path between two points is a straight line. If you travel from the origin along vector \(x\), and then along vector \(y\), the total distance \(||x|| + ||y||\) is always greater than or equal to the direct path \(||x+y||\).

3. Orthogonality (Perpendicularity)

Once we have length, the next most important geometric concept is the angle—specifically, the 90-degree angle.

The Definition:

Vectors \(x\) and \(y\) in an inner product space \(V\) are orthogonal (perpendicular) if:

$$
\langle x, y \rangle = 0
$$

A subset \(S\) is called an orthogonal set if every distinct pair of vectors in it is orthogonal. If those vectors also have a length of 1 (unit vectors), the set is called orthonormal.

A Simple Example:

Let’s go back to \(\mathbb{R}^2\).

Let \(x = (1, 1)\) and \(y = (1, -1)\).

Compute the inner product (dot product):

$$
\langle x, y \rangle = (1)(1) + (1)(-1) = 1 – 1 = 0
$$

Since their inner product is zero, these two vectors are orthogonal.

References

The theorem numbering in this post follows *Linear Algebra* (4th Edition) by Friedberg, Insel, and Spence. Some explanations and details here differ from the book.