18.4. SUBSPACES SPANS AND BASES 399

sum of two vectors in the set. However, multiplication by a negative scalar does not take avector in the set to another in the set.

not subspace not subspacesubspace not subspace

Observe how the above definition indicates that the claims posted on the picture arevalid. Now here are the two main examples of subspaces.

Theorem 18.4.5 Let A be an m×n matrix. Then Im(A) is a subspace of Fm. Alsolet

ker(A)≡ N (A)≡ {x ∈ Fn such that Ax= 0}

Then ker(A) is a subspace of Fn.

Proof: Suppose Axi is in Im(A) and a,b are scalars. Does it follow that aAx1 +bAx2is in Im(A)? The answer is yes because

aAx1 +bAx2 = A(ax1 +bx2) ∈ Im(A)

this because of the above properties of matrix multiplication. Note that A0 = 0 so 0 ∈Im(A) and so Im(A) ̸= /0.

Now suppose x,y are both in N (A) and a,b are scalars. Does it follow that ax+by ∈N (A)? The answer is yes because

A(ax+by) = aAx+bAy = a0+b0 = 0.

Thus the condition is satisfied. Of course N (A) ̸= /0 because A0 = 0.Subspaces are exactly those subsets of Fn which are themselves vector spaces. Recall

that a vector space is something which satisfies the vector space axioms on Page 273.

Proposition 18.4.6 Let V be a nonempty collection of vectors in Fn. Then V is a sub-space if and only if V is itself a vector space having the same operations as those definedon Fn.

Proof: Suppose first that V is a subspace. It is obvious all the algebraic laws hold on Vbecause it is a subset of Fn and they hold on Fn. Thus u+v = v+u along with the otheraxioms. Does V contain 0? Yes because it contains 0u= 0. Are the operations definedon V ? That is, when you add vectors of V do you get a vector in V ? When you multiply avector in V by a scalar, do you get a vector in V ? Yes. This is contained in the definition.Does every vector in V have an additive inverse? Yes because − v = (−1)v which is givento be in V provided v ∈V .

Next suppose V is a vector space. Then by definition, it is closed with respect to linearcombinations. Hence it is a subspace.