EXAM 1 Linear Algebra True or False

A transform like T(x1,x2,x3)=(1,x2,x3) is linear.

False. Linear transforms must satisfy T(cu)=cT(u) for ALL c, including c=0; a constant term like the '1' breaks this (T(0)≠0 here).

If A is n×n and Ax=b is consistent for every b in R^n, every column of A must be pivotal.

True. Consistent for every b means a pivot in every row; since A is square, that forces a pivot in every column too.

If three vectors are linearly dependent, any two of them must also be dependent.

False. The whole set can be dependent (e.g. one vector = sum of the other two) while any two of them, taken alone, are still independent.

If a linear system has more unknowns than equations, it must have infinitely many solutions.

False. It could still be inconsistent (e.g. two parallel planes) and have no solution at all.

If matrices A and B are row equivalent, they must have the same RREF.

True. Row equivalence means they reduce to the identical RREF.

If A is 2×3, the transformation x→Ax can never be one-to-one.

True. More columns than rows means there must be a free variable, so Ax=0 has nontrivial solutions — not one-to-one.

If an m×n matrix has linearly dependent columns, Ax=0 must have nontrivial solutions.

True. Dependent columns mean at least one free variable exists.

If b is not a linear combination of A's columns, Ax=b must be inconsistent.

True. That's the definition of b not being in the span of the columns.

If Ax=b is consistent, every column of A must be pivotal.

False. Consistency only requires b to be in the span of the columns — it says nothing about whether every column has a pivot (there could be free variables and still be consistent).

If Ax=0 has only the trivial solution, the columns of A must be linearly independent.

True. That's exactly the definition of independence.