For each statement that follows, answer true if the statement is always true and
ID: 3011811 • Letter: F
Question
For each statement that follows, answer true if the statement is always true and false otherwise. In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to show that the statement is not always true. Let L: R^n rightarrow^R^n be a linear operator. If L(x_1) = L(X_2), then the vectors x_1 and x_2 must be equal. If L_1 and L_2 are both linear operators on a vector space V, then L_1 + L_2 is also a linear operator on V, where L_1 + L_2 is the mapping defined by (L_1 + L_2)(V) = L_1 (v) + L_2(V) for all v elementof V If L: V rightarrow V is a linear operator and x elementof ker(L), then L(v + x) = L(v) for all V elementof V. Let L: R^2 rightarrow R^2 be a linear operator, and let A be the standard matrix representation of l. If L^2 is defined by L^2(x) = L(L(x)) for all x elementof R^2 then L^2 is a linear operator and its standard matrix representation is A^2.Explanation / Answer
1) False.
Reason : For any x in Rn , let x = ( a1 , a2 , ..... , an)
Define L: Rn -> Rn as L(x) = a1 + a2 + ..... + an
then L((1,0,0,....)) = L((0,1,0,.....)) but (1,0,0,....) (0,1,0,....)
2) true
Proof: Given (L1 + L2)(v) = L1 (v) + L2(v) for all v in V
to show L1 + L2 is a linear operator,
i) consider (L1 + L2)(v+w) = L1 (v+w) + L2(v+w) , for v,w in V ...[using given definition of (L1 + L2]
= L1 (v) + L1 (w) + L2(v) + L2(w) ...[ since L1 & L2 are linear operators]
= (L1 + L2)(v) + (L1 + L2)(w)
ii) consider (L1 + L2)(av) = L1 (av) + L2(av) for some scalar a ...[using given definition of (L1 + L2]
= a L1 (v) + aL2(v) ...[ since L1 & L2 are linear operators]
= a (L1 + L2)(v)
hence proved.
3) true
L(v+x) = L(v) + L(x)
= L(v) ...[ since x is in Ker(L) , hence L(x) = 0]
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