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14. Transition matrices are :
(A) Not at allinvertible. (8) Invertible always.
(C) Invertible sometimes. (D) Datanot complete.
15. Let A be any matrix. Then :
(A) rank (A) = rank (A‘), (B) rank (A) 2 rank (൧൯.
(0) rank (A) < rank (A*), (D) rank (A) > rank (A®)..

16. If Aisanm xn matrix, then:
(A) The null space of A and the row space of A are orthogonal complements in R”.
(B) The null space of A? and the column space of A are orthogonal complements in R™.
(C) Both (A) and (B) are correct.
(D) Neither (A) nor (B) are correct.

17. Let Ais ann x n matrix. The eigenspace of A corresponding to ) is same as:
(A) The null space of the matrix AI—A.
(B) The kernel of the matrix operator T,;_,4 :R” > R”
(C) The set of vectors for which Ax = ) ೫.
(D) All the above.
18. Let Ais ann xn matrix and suppose A has rank 7. Then :
(A) 17, is not one-to-one. (B) 2. = 015 101 an eigenvalue of A.
(C) The range of T, is {0}. (D) The kernel of T, is R®.
19. Which of the following is true ?
(ಹ) (u,v+w)=(u,v) + (४, ८). ൯) (५, ‏ن‎ + छ) = (४, ८) + (८, ४)

(C) Both (A) and (B) are true. (D) Neither (A) nor (B) is true.

20. Find the correct one from the given statements :

(A) Ifw is orthogonal to every vector of a subspace W, then u = 0.
(8) Ifw andv are orthogonal, then |(u, v)|=||x|| |||
(C) Ifw andv are orthogonal then || w+v||= || ||

(D) None of these.

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