235HW6 - Math 235: Linear Algebra HW 6 Problem 3.2.3...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 235: Linear Algebra HW 6 Problem 3.2.3 Problem 3.2.3 Let T be the linear operator on R 3 defined by T ( x 1 ,x 2 ,x 3 ) = (3 x 1 ,x 1- x 2 , 2 x 1 + x 2 + x 3 ) Is T invertible? If so, find a rule for T- 1 like the one which defines T . Yes it is invertible, and the inverse is given by T- 1 ( x 1 ,x 2 ,x 3 ) = (1 / 3 x 1 , 1 / 3 x 1- x 2 ,- x 1 ) Problem 3.2.4 For the linear operator T of Excercise 3, prove that ( T 2- I ) ( T- 3 I ) = 0 We start by computing ( T- 3 I )( x,y,z ): ( T- 3 I )( x,y,z ) = (3 x,x- y, 2 x + y + z )- (3 x, 3 y, 3 z ) = (0 ,x- 4 y, 2 x + y- 2 z ) We now plug this into ( T 2- I ) : ( T 2- I ) (0 ,x- 4 y, 2 x + y- 2 z ) = T ( T (0 ,x- 4 y, 2 x + y- 2 z ))- (0 ,x- 4 y, 2 x + y- 2 z ) = T (0 ,- x + 4 y, 3 x- 3 y- 2 z )- (0 ,x- 4 y, 2 x + y- 2 z ) = (0 ,x- 4 y, 2 x + y- 2 z )- (0 ,x- 4 y, 2 x + y- 2 z ) = (0 , , 0). Problem 3.2.8 Let V be a vectors space over a field F and T a linear operator on V . If T 2 = 0, whac can you say about the relation of the range of T to the null space of T ? Give an example of a linear operator T on R 2 such that T 2 = 0 but T 6 = 0. Since T 2 = 0 we know that T ( T ( v )) = 0 for each v V . So this means that T ( v ) is in the null space of V . Clearly T ( v ) is also in the range of V . This shows us that the range of V is contained in the null space of V . Let T be the linear operator on R 2 defined b T ( x,y ) = ( y, 0). T 2 ( a,b ) = T ( T ( a,b )) = T ( b, 0) = (0 , 0). So T 2 = 0. Also T (1 , 1) = (1 , 0) so T 6 = 0. Page 1 of 5 Math 235: Linear Algebra HW 6 Problem 3.2.11 Problem 3.2.11 Let V be a finite-dimensional vector space and let T be a linear operator on V . Suppose that rank ( T 2 ) =rank( T )....
View Full Document

Page1 / 5

235HW6 - Math 235: Linear Algebra HW 6 Problem 3.2.3...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online