# Linear Algebra Chapter 1 True Or False

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question
A homogeneous system of equations can be inconsistent.
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.
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.
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.
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.
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.
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The equation Ax=b is referred to as a vector equation.
False. The equation Ax=b is referred to as a matrix equation because A is a matrix.
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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.
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.