554-HW-3q - 3 Let V be an n dimensional vectorspace Let T V → V be a linear tranformation such that T V = null T Show n is even 3 Let V be a

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Unformatted text preview: 3) Let V be an n dimensional vectorspace. Let T : V → V be a linear tranformation such that T ( V ) = null T . Show n is even. 3) Let V be a vectorspace and T : V → V a linear transformation. Show that “ T ( V ) ∩ null T = { } ” ⇐⇒ “ T ( Tα ) = ⇒ Tα = 0”. 3) Let V be a vectorspace and T a linear operator on V . If T 2 = 0 but T 6 = 0 what can we say about the relationship of the range of T to the nullspace of T ? 3) Let V be a vectorspace with dim V = n and let T be a linear operator on V such that rank T = rank T 2 . Show that T ( V ) ∩ null T = { } . 3) Let T ∈ L ( F n ,F n ), let A be the matrix of T in the standard ordered basis for F n , and let W be the subspace spanned by the column vectors of A . 3) Find a basis for the range and the null space of A = 1 2 1 1 1- 1 3 4 3) T ( x,y ) = (- y,x ) (a) The matrix for T in the standard ordered basis for R 2 is given by Te 1 = (0 , 1) = 0 e 1 + 1 e 2 ; Te 2 = (- 1 , 0) = (- 1) e 1 + 0 e 2 , i.e.[ T ] B = 1- 1 0 (b) Using the new ordered basis B = { (1 , 2);(1 ,- 1) } we have Te 1 = (2 , 1) = 1 e 1 + 1 e 2 ; Te 2 = (- 1 , 1) = 0 e 1 + (- 1) e 2 , i.e.[ T ] B = 1 1- 1 (c) Prove that for every real number c the linear operator ( T- cI ) is invertible. ( T- cI )( x,y ) = T ( x,y )- c ( x,y ) = (- y,x )- ( cx,cy ) = (- cx- y,x- cy ) To compute [( T- cI )] B , ( T- cI ) e 1 = (- c, 1) = (- c ) e 1 + 1 e 2 ;( T- cI ) e 2 = (- 1 ,- c ) = (- 1) e 1 + (- c ) e 2 , i.e.[( T- cI )] B =- c 1- 1- c . But the matrix M = 1 c 2 +1- c- 1 1- c can be seen to be the inverse of [( T- cI )] B by multiplying them out, so the linear operator corresponding to...
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554-HW-3q - 3 Let V be an n dimensional vectorspace Let T V → V be a linear tranformation such that T V = null T Show n is even 3 Let V be a

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