Math 110 - Fall 2002 - Geba - Midterm 1

Math 110 - Fall 2002 - Geba - Midterm 1 - 1

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Unformatted text preview: 05/12/2003 MON 15:45 FAX 6434330 MOFFITT LIBRARY I001 MAT 110 — MIDTERM 9/27/0Jt D'Gelibfb 1. Let T : R3 —> R2 the linear transformation defined by T(x,y,2) = ($+y+z,$+3y+5z). a) Find N (T), R(T) and compute dim N (T), dim R(T). b) Let ,8 and 7 the standard bases for R3 and R2 respectively. Consider also 0: — {(1, 1, 1), (2,3,4), (3, 4, 6)} basis for R3. Compute Q the change of coordinate matrix from fl to at and the represen— tation matrices [T]g,[T]g. Check that mrc=nn 2. Let V and W two vector spaces over the rational numbers field Q and T : V —> W which satisfies T(m+y) = T(:c) + T(y). Prove that T is a linear transformation. 3. Let m and 71 two positive integers. Construct a linear transformation T such that nullity(T) : m and rank(T) : n. 4. Let T : V % V a linear transformation, Where V is a finite—dimensional vector space. Prove that if r'ank(T):rank(T2) then R(T) fl N(T) : {O}. 5. Let A,B two square matrices, A,B E Jllnxn(R) such that In 7 AB is invertible. Prove that In — BA is invertible. ...
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