7.x + 3y -52 +8w =-3
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.