Vector spaces

Concepts

• Vector in ℝⁿ 10 exercisesA vector in ℝ^n is an ordered list of n real numbers, which geometrically looks like an arrow from the origin.
• Vector space 10 exercisesA real vector space is a set V with an addition and a scalar multiplication by reals satisfying eight axioms; its elements are ca…
• Vector subspace 10 exercisesA vector subspace W of a vector space V is a nonempty subset that is itself a vector space under the same operations.
• Linear combination and span 10 exercisesA linear combination of several vectors is a sum of scalar multiples of them; the span is the set of all possible linear combinat…
• Linear independence 10 exercisesA set of vectors is linearly independent if the only linear combination of them that gives the zero vector is the one with all co…
• Basis and dimension 10 exercisesA basis mathcal B of a vector space V is a linearly independent set that spans it; its number of vectors is the dimension of V.
• Rank of a matrix 10 exercisesThe rank of a matrix A is the dimension of the subspace spanned by its columns.