1111: Linear Algebra I
Dr. Vladimir Dotsenko (Vlad) Lecture 20
Linear maps and change of coordinates
As a last step, let us exhibit how matrices of linear maps transform under changes of coordinates.
Lemma 1. Letϕ:V→Wbe a linear operator, and suppose thate1, . . . , en ande10, . . . , en0 are two bases of V, andf1, . . . , fm andf10, . . . , fm0 are two bases ofW. Then
Aϕ,e0,f0 =Mf0,fAϕ,e,fMe,e0 =M−1f,f0Aϕ,e,fMe,e0.
Proof. Let us take a vectorv∈V. On the one hand, the formula of Lemma 2 tells us that (ϕ(v))f0=Aϕ,e0,f0ve0.
On the other hand, applying various results we proved earlier, we have
(ϕ(v))f0 =Mf0,f(ϕ(v)f) =Mf0,f(Aϕ,e,fve) =Mf0,f(Aϕ,e,f(Me,e0ve0)) = (Mf0,fAϕ,e,fMe,e0)ve0.
Therefore,
Aϕ,e0,f0ve0 = (Mf0,fAϕ,e,fMe,e0)ve0
for every ve0. From our previous classes we know that knowingAvfor all vectorsvcompletely determines the matrixA, so
Aϕ,e0,f0 = (Mf0,fAϕ,e,fMe,e0) = (M−1f,f0Aϕ,e,fMe,e0) because of properties of transition matrices proved earlier.
Remark 1. Our formula
Aϕ,e0,f0 =Mf0,fAϕ,e,fMe,e0.
shows that changing from the coordinate systemse,f tosomeother coordinate system amounts to multiply- ing the matrixAϕ,e,f by some invertible matrices on the left and on the right, so effectively to performing a certain number of elementary row and column operations on this matrix. This is very useful (but not applicable to a more narrow class of linear transformations, see below).
Remark 2. A linear operatorϕ:V→Vis often called alinear transformation. For a linear transformation, it makes sense to use the same coordinate system for the input and the output. By definition, the matrix of a linear operatorϕ: V→V relative to the basise1, . . . , en is
Aϕ,e:=Aϕ,e,e.
Lemma 2. For a linear transformationϕ:V→V, and two basese1, . . . , en ande10, . . . , en0 of V, we have Aϕ,e0 =M−1e,e0Aϕ,eMe,e0.
Proof. This is a particular case of Lemma 1.
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Remark 3. Proposition 5 shows that for a square matrixA, the change A7→C−1AC with an invertible matrix C, corresponds to the situation where A is viewed as a matrix of a linear transformation, and C is viewed as a transition matrix for a coordinate change. You verified in your earlier home assignments that tr(C−1AC) =tr(A) and det(C−1AC) =det(A); these properties imply that the trace and the deter- minant do not depend on the choice of coordinates, and hence reflect some geometric properties of a linear transformation. (In case of the determinant, those properties have been hinted at in our previous classes:
determinants compute how a linear transformation changes volumes of solids).
Examples of linear maps and coordinate changes
Example 1. As we know, every linear mapϕ: Rn →Rk is given by ak×n-matrixA, so thatϕ(x) =Ax.
Example 2. LetVbe the vector space of all polynomials in one variablex. Consider the functionϕ:V→V that maps every polynomialf(x)toxf(x). This is a linear map:
x(f1(x) +f2(x)) =xf1(x) +xf2(x), x(cf(x)) =c(xf(x)).
Example 3. LetVbe the vector space of all polynomials in one variablex. Consider the functionψ:V→V that maps every polynomialf(x)tof0(x). This is a linear map:
(f1(x) +f2(x))0=f10(x) +f20(x), (cf(x))0=cf0(x).
Example 4. LetVbe the vector space of all polynomials in one variablex. Consider the functionα:V→V that maps every polynomial f(x)to 3f(x)f0(x). This is not a linear map; for example, 17→0, x7→3x, but x+17→3(x+1) =3x+36=3x+0.
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