Vectors and matrices
Learning objectives
- Distinguish scalars, vectors, and matrices
- state dimensions
- compute vector length and matrix-vector products.
Concept and explanation
A vector in R^n is an ordered list; an m×n matrix represents m linear combinations of n inputs. Addition requires equal dimensions. Ax is defined only when A has n columns and x has n entries. The Euclidean norm is ||x||₂=sqrt(sum x_i²).
Worked examples
Worked example 1
x=(3,−4) has ||x||₂=5.
Worked example 2
A=[[1,2],[−1,3]], x=(2,1) gives Ax=(4,1); dimension check (2×2)(2×1)=(2×1).
Common mistake and counterexample
Multiplying corresponding entries is not matrix-vector multiplication; use row dot products.
Knowledge check and mastery evidence
4/4: classify objects, state dimensions, compute the norm, and calculate Ax; compatibility is mandatory. Connects to transformations and robotics.
Prerequisites
Algebra, coordinates and systems notation.
Key terms
- scalar
- vector
- matrix
- dimension
- norm
Notation and definitions
x∈Rⁿ; A∈Rᵐˣⁿ; Ax; ||x||₂
A · Interactive foundation practice
Compute ||(1,2,2)||₂.
B · Application
Apply the current linear algebra concept to robotics, data, control, simulation or another meaningful linear model. Define variables, dimensions, assumptions and units.
C · Reasoning and error detection
Diagnose a topic-specific error, explain why it fails, correct it and verify the correction independently.
D · Challenge / transfer
Connect symbolic, graphical and numerical representations in a less-structured problem; state assumptions and limitations.
Check and summary
Connect the symbolic, graphical, numerical and verbal representations; state units, one likely error and an independent validation method.
Next class: Matrix operations