math 311  GLOSSARY (Linear Algebra)  E. L. Lady

Let v_{1}, …, v_{p}, w be vectors in ℝ^{m}. Let A be the m × p matrix whose columns are v_{1}, …, v_{p}:

A = [ v_{1} … v_{p}]

▶ w belongs to the subspace spanned by v_{1}, …, v_{p}.

means

▶ w is a linear combination of v_{1}, …, v_{p}.

means

▶ There exist scalars x_{1}, …, x_{p} such that w = x_{1}v_{1} + ⋯ + x_{p}v_{p}. means

▶ There exists a vector [ x_{1} ; ⋮ ; x_{p} ] in ℝ^{p} such that A [ x_{1} ; ⋮ ; x_{p} ] = w.

means

▶ The linear system Ax = w has a solution.

means

▶ The linear system Ax = w is consistent.

▶ v_{1}, …, v_{p} are linearly independent.

means

▶ If x_{1}v_{1} + ⋯ + x_{p}v_{p} = 0 then x_{1} = 0, …, x_{p} = 0. means

▶ If A [ x_{1} ; ⋮ ; x_{p} ] = [ 0 ⋮ 0 ] then [ x_{1} ; ⋮ ; x_{p} ] = 0.

means

▶ The homogeneous system Ax = 0 has only the trivial solution. math 311 BRIDE OF GLOSSARY

Any linear transformation L: ℝ^{m} → ℝ^{n} is given by a formula of the form L(x) = Ax, where A is an m × n matrix. More specifically, the columns of A are L(v_{1}), …, L(v_{n}) where v_{i} is the column vector with 1 in the ith coordinate and 0 in the other coordinates. If L and A are related in this way, then

Ker L

is the same as

The null space of A

is the same as

The solution space to the homogeneous system Ax = 0.

▶ w ∈ Range L

means

▶ w = L(x) for some x ∈ ℝ^{n}

means

▶ The linear system Ax = w has a solution

means

▶ w belongs to the column space of A.

▶ L is one-to-one.

means

▶ If L(v) = L(v′) then v = v′.

means

▶ For each w ∈ ℝ^{m} there is at most one x ∈ ℝ^{n} such that L(x) = w.

means

▶  If w ∈ ℝ^{m} then the linear system Ax = w has either a unique solution or no solution at all.