19.--_ : If +5 is a factor of the characteristic polynomial of A, then 5 s an ei
ID: 3117071 • Letter: 1
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19.--_ : If +5 is a factor of the characteristic polynomial of A, then 5 s an eigenvalue of A. 20. : v-v = IMI2 for any vector v E Rn 21. : For any scalar c, u. (cv) = c(u, v) 22. For a square matrix A, any vector in the row space of A is orthogonal to any vector in the Null(A). 23._ If B is orthogonal basis for a subspace w and c 0 is a scalar then multiplying a vector of B by c will result in a new orthogonal basis. 24. ·u-v-v-u=0for any vectors u,v E Rn 25· : For any scalar c, lev-chvil 26. : If B is a basis for a subspace w and is orthogonal to every vector of B then x E wi 27. : If W a subspace of Rn and v is a nonzero vector in W, then it is possible that v can also be in 28. : If ATx = 0, then x Col(A). 29. If w is any subspace of R3, then dim (W) + dim(W*) = 3. 30.: It is possible for u to not be orthogonal to v and not be orthogonal to w, but be orth (v+w)Explanation / Answer
19. The statement is False. If +5 is a factor in the characteristic polynomial of A, then -5 (and not 5)is an eigenvalue of A.
20. The statement is True. v.v is the square of the nporm of v.
21. The statement is True.
22. The statement is False. Any vector in col(A) is orthogonal to any vector in Null(A).
23. The statement is True. If u.v = 0 ,then cu.v = 0.
24. The statement is True. Inner product of vectors is commutative.
25. The statement is true.
26. The statement is True. It is the definitionof orthogonal complement.
27. The statement is False as W W = {0}.
28. The statement is True. If ATx = 0, then x is in Null(AT). Also, N(AT) = C(A).
29. The statement is True.
30. The statement is True. If w = -v , thenv+w = 0 so that u.(v+w) = 0 even if u.v 0 and u.w0
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