5.3 The 2-norm of a Vector

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Properties: 5.4 Algebraic Properties of the 2-norm of a Vector
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The 2-norm of a Vector

Let $n \in N$ be a positive integer and let $\vec{x} \in R^{n}$ . Then, the 2-norm $\vec{x} \in R^{n}$ is given by the square root of the sum of the square of the elements.

| | \vec{x} | |_{2} = \sqrt{x_{1}^{2} + x_{2}^{2} + \dots + x_{n}^{2}} = \sqrt{\sum_{j = 1}^{n} x_{j}^{2}}

Another way to calculate the 2-norm is to take the 5.1 Inner Product Between Vectors using the following formula:

| | \vec{x} | |_{2} = \sqrt{\vec{x} \cdot \vec{x}}

Remark. The two-norm is also known as the euclidean norm.
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Remark. From this point on, the two norm will be denoted with the following notation:

| | \vec{x} | | = | | \vec{x} | |_{2}

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Remark. There are many other vector-norms two consider. However, we will focus on the 2-norm as it is the most important when it comes to introductory linear algebra.
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Remark. In Multivariable Calculus, the 2-norm of a Vector is also known as the magnitude.
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Remark. The two-norm is also known as the euclidean norm. ⧫

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Remark. There are many other vector-norms two consider. However, we will focus on the 2-norm as it is the most important when it comes to introductory linear algebra. ⧫

Remark. The two-norm is also known as the euclidean norm.
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Remark. There are many other vector-norms two consider. However, we will focus on the 2-norm as it is the most important when it comes to introductory linear algebra.
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