Norms of Vectors and Matrices

Key idea

Measures of size such as ℓ¹, ℓ², ℓ∞ and Frobenius.

In depth

Norms measure size: the 1-norm (sum of absolute values), 2-norm (Euclidean length), and ∞-norm (max entry). For matrices, ‖A‖ gives the maximal stretching of a vector. Norms must satisfy the triangle inequality.

Worked example

For (3, −4): ‖·‖₁ = 7, ‖·‖₂ = 5, ‖·‖∞ = 4.

Formula

Real-world uses

• Measuring error size, distance and stability of models in machine learning

Common mistake

Assuming all norms give the same size; they differ, but are equivalent in finite dimensions.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Linear Algebra

• Triple Product – Scalar triple product gives parallelepiped volume.
• Homogeneous Systems – Ax = 0 always has the trivial solution.
• Cofactor Expansion – Compute determinants recursively.
• Properties of Determinants – Multiplicative; row swaps change sign.
• Geometric Meaning of Determinants – Area or volume scaling factor.
• Cramer’s Rule for Systems – Solve systems using determinants.