Multivariable Advanced Calculus

30 BASIC LINEAR ALGEBRAProof: This follows immediately from the proof of Theorem 3.2.10. You do exactlythe same argument except you start with {v 1 , · · · , v r } rather than {v 1 }.It is also true that any spanning set of vectors can be restricted to obtain a basis.Theorem 3.2.12 Let V be a subspace of Y, a finite dimensional vector space ofdimension n and suppose span (u 1 · · · , u p ) = V where the u i are nonzero vectors. Thenthere exist vectors, {v 1 · · · , v r } such that {v 1 · · · , v r } ⊆ {u 1 · · · , u p } and {v 1 · · · , v r }is a basis for V .Proof: Let r be the smallest positive integer with the property that for some set,{v 1 · · · , v r } ⊆ {u 1 · · · , u p } ,span (v 1 · · · , v r ) = V.Then r ≤ p and it must be the case that {v 1 · · · , v r } is linearly independent because ifit were not so, one of the vectors, say v k would be a linear combination of the others.But then you could delete this vector from {v 1 · · · , v r } and the resulting list of r − 1vectors would still span V contrary to the definition of r. This proves the theorem. 3.3 Linear TransformationsIn calculus of many variables one studies functions of many variables and what is meantby their derivatives or integrals, etc. The simplest kind of function of many variables isa linear transformation. You have to begin with the simple things if you expect to makesense of the harder things. The following is the definition of a linear transformation.Definition 3.3.1 Let V and W be two finite dimensional vector spaces. A function,L which maps V to W is called a linear transformation and written as L ∈ L (V, W )if for all scalars α and β, and vectors v, w,L (αv+βw) = αL (v) + βL (w) .An example of a linear transformation is familiar matrix multiplication, familiar ifyou have had a linear algebra course. Let A = (a ij ) be an m × n matrix. Then anexample of a linear transformation L : F n → F m is given byHere(Lv) i≡⎛⎜v ≡ ⎝v 1.v nn∑a ij v j .j=1⎞⎟⎠ ∈ F n .In the general case, the space of linear transformations is itself a vector space. Thiswill be discussed next.Definition 3.3.2 Given L, M ∈ L (V, W ) define a new element of L (V, W ) ,denoted by L + M according to the rule(L + M) v ≡ Lv + Mv.For α a scalar and L ∈ L (V, W ) , define αL ∈ L (V, W ) byαL (v) ≡ α (Lv) .