Question

. True or false?

(1) There is at most one pivot in any row.

(2) There is at most one pivot in any column.

(3) There is at least one pivot in any row.

(4) There is at least one pivot in any column.

(5) There cannot be more free variables than pivot variables.

(6) There is a linear system that has exactly two solutions.

(7) The weights c1, ..., cp ? R in a linear combination c1v1 + ... + cpvp cannot all be zero.

(8) Given nonzero vectors u, v in R n , span{u, v} contains the line through u and the origin. Hint: can you describe this line as a set of vectors?

(9) Asking whether the linear system corresponding to a1 a2 a3 b is consistent, is the same as asking whether b is a linear combination of a1, a2, a3.

(10) If the augmented matrix of a linear system has two identical rows, the linear system is necessarily inconsistent

Answer 1

(1) True. If a matrix is in row-echelon form, then the first non-zero entry of each row is called a pivot. Obviously, there can be only one pivot in a row.

(2).True. In the row- echelon form there is only a single non-zero entry in a column with a pivot. Therefore there is at most one pivot in each column.

(3). False. In a row of zeros, there is no pivot.

(4). False. In a column of zeros, there is no pivot.

(5). False. In the equation x+y+z = 0, the variable x is the pivot variable while y and z are free variables.

(6). False. A linear system can either have a unique solution or infinite solutions or no soluttion. It cannot have exactly 2 solutions.

(7).False. The weights are all zero if the v₁,v₂,…,v_p are linearly independent.

(8).True. Span{u,v} is the plane containing both the vectors u and v. Therefore, it contains the line through the origin and u.

(9) True. This is as per the definition of a linear combination.

(10). False. The linear system x+y+z = 0, x+y+z = 0 has infinite solutions.