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HW 3 4 3 (1) Consider the linear transformation T :R !R de ned by T (x; y; z; w) = (x + 2yz +w;x 2y +z 2w; x + 2yz): (a) By directly using the de nition of the range of a linear transformation, write down a description of the rangeR(T ); and determine a nonzero vector in it. (b) Find description ofR(T ) as the intersection of hyperplanes and deduce a basis forR(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 theoremT :V !W , then dim(V ) =r(T )+n(T ) for the linear transformation considered above. 1

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Answered Same Day Dec 23, 2021

Solution

David answered on Dec 23 2021
128 Votes
1. Range is defined as
R(T ) := {(x+2y−z+w,−x−2y+z−2w, x+2y−z) ∈ R3 : x, y, z, w ∈ R}
Let y = z = w = 0 and x = 1 then we have
x + 2y − z + w = 1
−x− 2y + z − 2w = −1
x + 2y − z = 1
Hence we see that (1,−1, 1) ∈ R(T ) is a non zero element.
2. In R4, define the hyperlane H1, H2 and H3 by:
H1 := {(x, y, z) : x− y + z = 0}
H2 := {(x, y, z) : 2x− 2y + 2z = 0}
H3 := {(x, y, z) : −x + y − z = 0}
H4 := {(x, y, z) : x− 2y =...
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