無年度 - 主題課程_行列式和線性方程式：線性方程式#107934

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.

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 following linear equation systems, express these equations as the matrix form Ax=b. Then find the solution of the vector x. (5%)