(I) The function T : M22→R defined by T(A) = det A is a linear transformation.(II) The composition of two linear transformations is also a lineartransformation.(III)T here is exactly one linear transformation T : V →W for which T(u + v) = T(u − v) for all vectors u and v in V.(IV) The inverse of a one-to-one linear transformation is a lineartransformation.A. II, III. IV True and I FalseB. II, III True and I , IV FalseC. I, III, IV True and II FalseD. II, . IV True and I, III False

(b) Assume that {v1, v2, · · · , vk} is a basis of Y . Show that the linear transform T : X → Y from (a) is invertible. (Hint: Use (a) to define a suitable linear transform U : Y → X and show that it is the inverse of T .)

et T and S be linear transformations on a finite dimensional vector space V, where S is non-singular. Then,ans.Rank (ST)= Rank(TS) ≠ Rank TRank (ST) ≠ Rank(TS) ≠ Rank TRank (ST) ≠ Rank(TS) = Rank TRank (ST) = Rank(TS) = Rank T Previous Marked for Review Next

Determine whether the function involving the n × n matrix A is a linear transformation.T: Mn,n → Mn,n, T(A) = A−1

Which of the following is/are true? (A) Every linear transformation from R 2 to R 2 maps lines onto points or lines (B) Every surjective linear transformation from R 2 to R 2 maps lines onto lines (C) Every bijective linear transformation from R 2 to R 2 maps pairs of parallel lines to pairs of parallel lines (D) Every bijective linear transformation from R 2 to R 2 maps pairs of perpendicular lines to pairs of perpendicular lines

Let T𝑇 and S𝑆 be linear transformations defined by T(x,y)=(2x−3y,−y)𝑇(𝑥,𝑦)=(2𝑥−3𝑦,−𝑦) and S(x,y)=(x−y,x)𝑆(𝑥,𝑦)=(𝑥−𝑦,𝑥). What is the matrix corresponding to the linear transformation T∘S𝑇∘𝑆?