question

A homogeneous system of equations can be inconsistent.

answer

False. A homogeneous equation can be written in the form Ax=0, where A is an m×n matrix and 0 is the zero vector in ℝm. Such a system Ax=0 always has at least one solution, namely x=0. Thus, a homogeneous system of equations cannot be inconsistent.

question

If x is a nontrivial solution of Ax=0, then every entry in x is nonzero.

answer

False. A nontrivial solution of
Ax=0 is a nonzero vector x that satisfies Ax=0. Thus, a nontrivial solution x can have some zero entries so long as not all of its entries are zero.

question

The effect of adding p to a vector is to move the vector in a direction parallel to p.

answer

True. Given v and p in ℝ2 or ℝ3, the effect of adding p to v is to move v in a direction parallel to the line through p and 0.

question

The equation Ax=b is homogeneous if the zero vector is a solution.

answer

True. A system of linear equations is said to be homogeneous if it can be written in the form Ax=0, where A is an m×n matrix and 0 is the zero vector in ℝm. If the zero vector is a solution, then b=Ax=A0=0.

question

If Ax=b is consistent, then the solution set of Ax=b is obtained by translating the solution set of Ax=0.

answer

True. Suppose the equation Ax=b is consistent for some given b, and let p be a solution. Then the solution set of Ax=b is the set of all vectors of the form w=p+vh, where vh is any solution of the homogeneous equation Ax=0.

question

The equation Ax=b is referred to as a vector equation.

answer

False. The equation Ax=b is referred to as a matrix equation because A is a matrix.

question

A vector b is a linear combination of the columns of a matrix A if and only if the equation Ax=b has at least one solution.

answer

True. The equation Ax=b has the same solution set as the equation x1a1+x2a2+•••+xnan=b.

question

The equation Ax=b is consistent if the augmented matrix
Ab has a pivot position in every row.

answer

False. If the augmented matrix Ab has a pivot position in every row, the equation equation Ax=b may or may not be consistent. One pivot position may be in the column representing b.

question

The first entry in the product Ax is a sum of products.

answer

True. The first entry in Ax is the sum of the products of corresponding entries in x and the first entry in each column of A.

question

If the columns of an m×n matrix A span ℝm, then the equation Ax=b is consistent for each b in ℝm.

answer

True. If the columns of A span ℝm, then the equation Ax=b has a solution for each b in ℝm.

question

If A is an m×n matrix and if the equation Ax=b is inconsistent for some b in ℝm, then A cannot have a pivot position in every row.

answer

True. If A is an m×n matrix and if the equation Ax=b is inconsistent for some b in ℝm, then the equation Ax=b has no solution for some b in ℝm.