Question

Is the Euclidean norm function $N(\mathbf{v})=\|\mathbf{v}\|$, for $\mathbf{v} \in \mathbb{R}^n$, linear?

    Is the Euclidean norm function $N(\mathbf{v})=\|\mathbf{v}\|$, for $\mathbf{v} \in \mathbb{R}^n$, linear?

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 7, Problem 11

Instant Answer

Step 1

For a vector $\mathbf{v} \in \mathbb{R}^n$ with components $v_1, v_2, \dots, v_n$, the Euclidean norm (or 2-norm) of $\mathbf{v}$ is given by: \[ N(\mathbf{v}) = \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + \dots + v_n^2}. \] Step 2: Recall the definition of a linear Show more…

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Is the Euclidean norm function $N(\mathbf{v})=\|\mathbf{v}\|$, for $\mathbf{v} \in \mathbb{R}^n$, linear?

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