Linear transformations and matrices

Span in linear algebra (algebraic interpretation)

The set of all possible vectors (points) that can be reached using a linear combination of those vectors.

Span in a vector space (geometric interpretation)

A physical region (subspace) comprising specific locations or coordinates within that space.

Name and describe the two ways that a vector can be conceptualised.

1. As a displacement: An arrow starting at the origin and pointing in a specific direction with a certain magnitude (length).
2. As a point: The coordinates of the arrowhead itself in a given coordinate system (e.g., (𝑥,𝑦) in 2D space).

Scaled vector

A vector that is produced when the initial vector is multiplied by a scalar (a real number). This operation, often called scalar multiplication, changes the magnitude (length) of the vector, while generally keeping its direction the same, unless the scalar is negative. 

Linear combination of vectors

The sum of scaled vectors:

c₁ * v₁ + c₂ * v₂ + c₃ * v₃ + ... + c_n * v_n