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MATB24H3 (60)

Sophie Chrysostomou (10)

Lecture

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MathematicsCourse Code

MATB24H3Professor

Sophie ChrysostomouThis

**preview**shows page 1. to view the full**5 pages of the document.**University of Toronto at Scarborough

Department of Computer & Mathematical Sciences

MAT B24S Fall 2011

Lecture 9

Suppose that T:V−→ V′is a linear transformation from vector space

Vwith ordered basis Binto the vector space V′with ordered basis B′. Let

[T]B, B′be the matrix representation of Trelative to Bamd B′. Now let

T′:V′−→ Wbe a linear transformation from the vector space V′to the

vector space Wwith ordered basis B′′ .Then [T′◦T]B,B′′ = [T′]B′, B′′ [T]B, B′

This follows from the fact that for all v∈Vand v′∈V′we have that

1. (T(v))B′= [T]B,B′vB

2. (T′(v′))B”= [T′]B′,B′′ v′

B′

3. let v′=T(v)

4. (T′◦T)(v)B”= (T′(T(v)))B”= [T′]B′B”(T(v))B′= [T′]B′,B′′ [T]B,B′vB

Now suppose that Band B′are two ordered bases of Vand T:V−→ V

is a linear transformation.

Suppose that CB, B′is the change of coordinate matrix from Bto B′.

Then for all v∈Vwe have vB′=CB, B′vB. Suppose also that CB′, B is

the change of coordinate matrix from B′to B. Then for all v∈Vwe have

vB=CB′, BvB′.

If [T]Bis the matrix rep of Trelative to Bonly i.e. T(v)B= [T]BvB

for all v∈V. and that [T]B′is the matrix rep of Trelative to B′only i.e.

T(v)B′= [T]B′vB′for all v∈V. Then we have that

[T]B=C−1

B, B′[T]B′CB, B′

Thus [T]B′and [T]Bare similar matrices.

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Remember that Dis a similar matrix to Aif Dcan be obtained from

Aby performing elementary row (or column) operations. Now we see that

they are also both matrix rep of the same linear transformation Trelative

to suitable diﬀerent ordered basis.

In the same way then it may be possible to ﬁnd the matrix rep of T rel

to some ordered basis such that this matrix is a diagonal.

Remember also that similar matrices have the same eigenvalues. (ie. λis

an eigenvalue of Aif Av=λvfor some eigenvector vof A) We say that a

lin trans Thas an eigenvalue λif T(v) = λvand we call van eigenvector of

T.

Review Chapter 5 from last year.

It is sometimes of interest and convenient to have the matrix rep of a

linear transformation relative to an ordered basis Bin a diagonal form. This

is only possible if a matrix rep of the linear transformation relative to a

known basis is diagonalizable.

Suppose we have Athe matrix rep of Trelative to some known basis

B′. We need to ﬁnd if it is possible to ﬁnd a diagonal matrix which is the

representation of Trelative to some basis Bwe need to ﬁnd out if Ais

diagonalizable.

Thus we need to do the following:

1. We ﬁnd the characteristic polynomial |A−λI|.

2. We ﬁnd the eigevalues of Awhich are the roots of the characteristic

polynomial of A.

3. Find the algebraic multiplicity of each of the eigenvalues.

4. For each eigenvalue λiﬁnd the eigenspace Eλiwhich is the nullspace of

A−λi.

5. The dim(Eλi) is the geometric multiplicity of λi.

6. If the geometric multiplicity of λi= the algebraic multiplicity of λifor

each λi, then A is diagonalizable!

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