Show that ||x|| is a norm on R^n

• Samuelb88
In summary, ||x|| = \sqrt{x \cdot x} is a norm on \mathbb{R}^n, satisfying the properties of being equal to 0 if and only if x=0, being scalar multiple invariant, and obeying the Cauchy-Schwarz inequality.
Samuelb88

Homework Statement

Show $||x|| = \sqrt{x \cdot x}$ is a norm on $\mathbb{R}^n$.

Homework Equations

Prop. 1. $||x|| = 0$ IFF $x=0$.
2. $\forall c \in \mathbb{R}$ $||cx|| = |c| \, ||x||$.
3. $||x+y|| \leq ||x|| + ||y||$.

Cauchy-Schwarz Inequality.

The Attempt at a Solution

Just want to check if I am showing this correctly. Note that I am using $x$ in place of $\vec{x} \in \mathbb{R}^n$.

Let $x \in \mathbb{R}^n$. Suppose that $||x|| = \sqrt{x \cdot x}$.

1. Suppose first that $||x||=0$. Then $0 = \sqrt{x \cdot x}$ imples $x \cdot x = 0$. But $x \cdot x = x_1^2 + \cdots + x_2^2 = 0$ if and only if $x_i =0$ $\forall x_i$. Thus if $||x|| =0$, then $x=0$. Suppose next that $x=0$. Then $||0|| = \sqrt{0^2 + \cdots + 0^2} = 0$. Therefore if $x=0$, then $||x||=0$.

2. Let $c \in \mathbb{R}$. Then $||cx|| = \sqrt{(cx_1)^2 + \cdots + (cx_n)^2} = c ||x||$.

3. This follows from the Cauchy-Schwarz inequality which I have proven and will not show here.

Everything looks correct.

1. What is a norm?

A norm is a mathematical concept that measures the length or size of a mathematical object, such as a vector. It is a way to quantify the distance between points in a mathematical space.

2. How is a norm defined on R^n?

A norm on R^n is typically defined as a function ||x|| that satisfies three properties: non-negativity, homogeneity, and the triangle inequality. These properties ensure that the norm measures the length of a vector in a consistent and meaningful way.

3. Why is ||x|| considered a norm on R^n?

||x|| is considered a norm on R^n because it satisfies all three properties mentioned above: non-negativity, homogeneity, and the triangle inequality. This means that ||x|| is a valid function for measuring the length of a vector in R^n.

4. Can you prove that ||x|| is a norm on R^n?

Yes, it is possible to prove that ||x|| is a norm on R^n. This can be done by showing that ||x|| satisfies all three properties of a norm: non-negativity, homogeneity, and the triangle inequality. This proof involves using mathematical logic and properties of vector spaces.

5. What are some real-world applications of ||x|| as a norm on R^n?

The concept of a norm and ||x|| specifically has many real-world applications in fields such as physics, engineering, and computer science. It is used to measure distances and sizes in mathematical spaces, which is essential in many scientific and technological fields.

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