Label the following statements as true or false. (a) Any n-linear function 8: M;

Chapter 4, Problem 1

(choose chapter or problem)

Label the following statements as true or false. (a) Any n-linear function 8: M;|X(F) > F is a linear transformation. (b) Any n-linear function 8: Mnxn ( F) * F Is a linear function of each row of an n x n matrix when the other n 1 rows are held fixed. (c) If 8: MnXfl(F) * F is an alternating n-linear function and the matrix A G M x n (F) has two identical rows, then 8(A) = 0. (d) If 8: MnXT,(F) > F is an alternating n-linear function and B is obtained from A G Mnxn (F) by interchanging two rows of A, then 8(B) = 8(A). (e) There is a unique alternating n-linear function 8: MnXn (F) > F. (f) The function 8: Mnxn(F) -* F defined by 8(A) = 0 for every A G Mnxn ( F) is an alternating //-linear function.

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