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ㆍLet A be an m✕ n real-valued matrix of rank r. and b be an m✕ 1 real-valued vector. Which of the following statement is/are true? (A). The equation Ax = b has non-trivial solutions if n ＞ r ＞ 1 and b is all-zero. (B). The equation Ax = b has solutions if b belongs to the column space of A. (C). The equation Ax = b has only one solution when r = m. (D). The dimension of the nullspace of A is O if the dimension of the row space of A is r. (E). None of the above is true.
Consider the linear equation Ax = b with A 
. Which of the following statements are true?
(A) Ifrank( A ) = m, then there exists at least one solution.
(B) If rank( A ) = n, then there exists exactly one solution.
(C) Ifrank( A ) = n, then the column vectors of A are lincarly indcpendent.
(D) Ifn ＞ m, then there exists at least one solution.
(E) Ifm ＞ n, then there exists at most one solution.
,Let A = 
and define a transformation T : R2 → R3 by T(x) = Ax. Which
1-1
of the following statements is/are true?
(A) The image of x = 
under T is
.
(B) There is exactly one x whose image under T is 
. (C)The vector b is in the range of T if b is the image of some x in R2. (D) The vector b is in the range of T if the system Ax = b is inconsistent.
(E) The vector 
is not in the range of T.
非選
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