Linear Algebra Chapter 1 True Or False

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.

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.

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.

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.

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.

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

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

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.

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.

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

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.