math110s-hw8

math110s-hw8 - MATH 110 LINEAR ALGEBRA SPRING 2007/08...

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Unformatted text preview: MATH 110: LINEAR ALGEBRA SPRING 2007/08 PROBLEM SET 8 V denotes a finite dimensional vector space. If T ∈ End( V ), we will write T 2 = T ◦T , T 3 = T ◦T ◦T , etc. We let O ∈ End( V ) and I ∈ End( V ) denote the zero and identity operators, ie. O ( v ) = V and I ( v ) = v for all v ∈ V . 1. Let V be a vector space over F and S , T ∈ End( V ). (a) Show that I - S ◦ T is injective iff I - T ◦ S is injective. (b) T is called nilpotent if T n = O for some n ∈ N . Show that if T is nilpotent, then I - T is bijective. What is ( I - T )- 1 ? 2. Let V be a vector space over R and T ∈ End( V ) be an involution, ie. T 2 = I . Define V + := { v ∈ V | T ( v ) = v } and V- := { v ∈ V | T ( v ) =- v } . (a) Show that V + and V- are subspaces of V . (b) Show that V + ⊕ V- = V. 3. Let A ∈ R m × n and b ∈ R m . (a) Show that nullsp( A > A ) = nullsp( A ) . (b) Show that colsp( A > A ) = colsp( A > ) ....
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This note was uploaded on 08/01/2008 for the course MATH 110 taught by Professor Gurevitch during the Spring '08 term at Berkeley.

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math110s-hw8 - MATH 110 LINEAR ALGEBRA SPRING 2007/08...

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