Which of the following formulas define norms on ℝ^3 ? (a) 𝐯=√(2 v1^2+v2^2+3 v3^2), (b) 𝐯=√(v1^2+

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Which of the following formulas define norms on ℝ^3 ? (a)...

Which of the following formulas define norms on $\mathbb{R}^3$ ? (a) $\|\mathbf{v}\|=\sqrt{2 v_1^2+v_2^2+3 v_3^2}$, (b) $\|\mathbf{v}\|=\sqrt{v_1^2+2 v_1 v_2+v_2^2+v_3^2}$ (c) $\|\mathbf{v}\|=\max \left\{\left|v_1\right|,\left|v_2\right|,\left|v_3\right|\right\}$, (d) $\|\mathbf{v}\|=\left|v_1-v_2\right|+\left|v_2-v_3\right|+\left|v_3-v_1\right|$, (e) $\|\mathbf{v}\|=\left|v_1\right|+\max \left\{\left|v_2\right|,\left|v_3\right|\right\}$.

Which of the following formulas define norms on $\mathbb{R}^3$ ?
(a) $\|\mathbf{v}\|=\sqrt{2 v_1^2+v_2^2+3 v_3^2}$,
(b) $\|\mathbf{v}\|=\sqrt{v_1^2+2 v_1 v_2+v_2^2+v_3^2}$
(c) $\|\mathbf{v}\|=\max \left\{\left|v_1\right|,\left|v_2\right|,\left|v_3\right|\right\}$,
(d) $\|\mathbf{v}\|=\left|v_1-v_2\right|+\left|v_2-v_3\right|+\left|v_3-v_1\right|$,
(e) $\|\mathbf{v}\|=\left|v_1\right|+\max \left\{\left|v_2\right|,\left|v_3\right|\right\}$.

Key Concepts

Maximum Norm

Also known as the supremum norm or infinity norm, the maximum norm defines the norm of a vector as the maximum absolute value among its components. This norm satisfies all the key properties required to be a norm and is widely used in various applications.

Quadratic Form

A quadratic form is an expression involving squared terms and can be used to define norms, especially when they arise from an inner product. For a quadratic form to define a norm, it must be positive definite, ensuring that the associated expression only vanishes at the zero vector.

Triangle Inequality

The triangle inequality states that for any two vectors, the norm of their sum is less than or equal to the sum of their individual norms. This property is fundamental to ensuring that the norm defines a valid metric on the space, reflecting the idea of distance.

Homogeneity

Also known as scalability, homogeneity requires that multiplying a vector by a scalar multiplies its norm by the absolute value of that scalar. This property guarantees that the norm respects the scaling of the vector space.

Positive Definiteness

This concept ensures that the norm of any vector is zero only when the vector itself is the zero vector, and it is strictly positive for all other vectors. This is critical for the norm to be a valid measure of length.

Norm

A norm is a function that assigns a non-negative length or size to each vector in a vector space and must satisfy three properties: positive definiteness (it is zero if and only if the vector is the zero vector), homogeneity (it scales linearly with the absolute value of a scalar), and the triangle inequality (the norm of the sum of two vectors is at most the sum of their norms).

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