Linear Algebra

Fall 2025

Every spanning set of a vector space ___ contain at least one basis. MUST / MIGHT / CANNOT

MUST.
You can remove dependent vectors from a spanning set until it’s minimal — that’s a basis.

A linearly independent set ____ be a basis.

MIGHT.
It must also span the space; independence alone isn’t enough.

A vector space with dimension n ____ contain n+1 linearly independent vectors.

CANNOT.
Dimension is the maximum size of a linearly independent set.

If a set spans V, any larger set ____ also span V.

MUST.
Adding vectors can’t destroy spanning

If dim(V) = dim(W), then a linear transformation T : V to W ____ be an isomophism

MIGHT.
Same dimension allows it, but T still needs to be one-to-one and onto.