Question

Find the 1, 2, and [infinity] norms for the following vectors?

Answer 1

The 1-norm is the sum of absolute values of vector components, the 2-norm is the square root of the sum of squares of components, and the infinity-norm is the maximum absolute value of components.

Step-by-step explanation:

The question involves calculating the 1-norm, 2-norm, and infinity-norm for vectors. These norms are measures of vector magnitude in linear algebra and are defined as follows:

• The 1-norm (also called the Manhattan norm or taxicab norm) of a vector is the sum of the absolute values of its components.
• The 2-norm (also called the Euclidean norm) of a vector is the square root of the sum of the squares of its components, which corresponds to the length of the vector in Euclidean space.
• The infinity-norm (also called the max norm) of a vector is the maximum absolute value of its components.

For a vector v with components v1, v2, ..., vn, these norms are calculated as:

1. ||v||1 = |v1| + |v2| + ... + |vn|
2. ||v||2 = sqrt(v12 + v22 + ... + vn2)
3. ||v||∞ = max(|v1|, |v2|, ..., |vn|)