**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. ...

View
Full Document

- Fall '08
- GUREVITCH
- Math