Page 138 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes 9.3 Summary

 One can identify the effect of linear transformation on the space and study its effects by
means of algebra of matrices.

 This way one has insight of the meaning of similar matrices.
2
3
 The linear transformation T for R to R .
9.4 Keywords

Degree Decreasing: The differentiation operator D maps V into V, since D is ‘degree’ decreasing.

Linear Transformation: The statement that B is similar to A means that on each n-dimensional
space over F the matrices A and B represent the same linear transformation in two (possibly)
different ordered basis.

Unique Linear Operator: A unique linear operator U which carries  onto ', defined by
U ' , j = 1,...,n
j j
9.5 Review Questions

2
1. Let T be the linear transformation on R defined by
T x ,x x ,x
1 2 2 1
(a) What is the matrix of T in the standard basis for R ?
2
(b) What is the matrix of T in the standard basis , where 1,2 and 1, 1 ?
1 2 1 2
2. Let 1 , 2 , 3 be the basis for V and let 1 1 2 2 , 2 1 2 3 , 3 2 – 3
3
(a) Prove 1 , 2 , 3 is a basis and express 1 , 2 , 3 as a linear combination of
, and .
1 2 3
(b) If T is defined by T , i 1,2,3,...
i i
find a matrix A which represents T relative to basis.

9.6 Further Readings

Books Kenneth Hoffman and Ray Kunze, Linear Algebra

I.N. Herstein, Topics in Algebra
Michael Artin, Algebra

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