http://www.4shared.com/office/-hwJR8jt/Elementary_Linear_Algebra__9th.html Document Preview: HW 34 3(1) Consider the linear transformation T :R !R dened byT (x; y; z; w) = (x + 2yz +w;x 2y +z...

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HW 3 (1) Consider the linear transformation T : R4 −→ R3 defined by T (x, y, z, w) = (x + 2y − z + w, −x− 2y + z − 2w, x + 2y − z). (a) By directly using the definition of the range of a linear transformation, write down a description of the range R(T ), and determine a nonzero vector in it. (b) Find description of R(T ) as the intersection of hyperplanes and deduce a basis for R(T ) and the rank r(T ). (c) Find a basis for ker(T ) and determine n(T ) the nullity of T. (d) Verify the the theorem T : V −→W , then dim(V ) = r(T )+n(T ) for the linear transformation considered above. 1