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math110s-hw7

# math110s-hw7 - MATH 110 LINEAR ALGEBRA SPRING 2007/08...

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MATH 110: LINEAR ALGEBRA SPRING 2007/08 PROBLEM SET 7 If T End( V ), we will write T 2 = T ◦ T , T 3 = T ◦ T ◦ T , etc. We let O V End( V ) and I V End( V ) denote the zero and identity operators, ie. O V ( v ) = 0 V and I V ( v ) = v for all v V . 1. Let A, B F n × n . Define the function T : F n × n F n × n by T ( X ) = AXB for all X F n × n . (a) Show that T ∈ End( F n × n ). (b) Show that T is invertible if and only if A and B are non-singular matrices. 2. Let V be a finite dimensional vector space and T ∈ End( V ). Let dim( V ) = n . Let v V be such that T n - 1 ( v ) 6 = 0 and T n ( v ) = 0 . (a) Show that the vectors v , T ( v ) , T 2 ( v ) , . . . , T n - 1 ( v ) form a basis for V . (b) Let B be the basis in (a). What is the matrix representation [ T ] B , B ? 3. Let V and W be finite-dimensional vector spaces over F . (a) Let T ∈ Hom( V, W ). Prove the following. (i) If T is injective, then dim( V ) dim( W ). (ii) If T is surjective, then dim( V ) dim( W ). (iii) If T is bijective, then dim( V ) = dim( W ).

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math110s-hw7 - MATH 110 LINEAR ALGEBRA SPRING 2007/08...

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