True or False

1). Elementary row operations on an augmented matrix never change the solution set of the
associated linear system
2). The echelon form of a matrix is unique.
3). Whenever a system has free variables, the solution set contains many solutions.
4). In some cases, a matrix may be row reduced to more than one matrix in reduced echelon
form, using different sequences of row operations.
5). A basic variable in a linear system is a variable that is corresponding to a pivot column in
the coefficient matrix.
6). The solution set of the linear system whose augmented matrix is [a 1 a 2 a3 | b] is the same as
the solution set of the equation x1 a 1 + x2 a2 + x3 a 3 = b.
7). The set Span{u, v} is always visualized as a plane through the origin.
8). Asking whether the linear system corresponding to an augmented matrix [a1 a 2 a 3 | b] has
solution amounts to asking whether b is in Span{a 1 , a2 , a3 }.
9). The 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.
10). The equation Ax = b is consistent for all possible b if the augmented matrix [A b] has a
pivot position in every row.
11). The equation Ax = b is consistent for all possible b if the augmented matrix [A b] has a
pivot position in every column.
12). If A is an m → n matrix whose columns do not span Rm , then the equation Ax = b is incon-
sistent for some b in Rm .
13). The homogeneous equation Ax = 0 has a nontrivial solution if and only if the equation has
at least one free variable.
14). If x is a nontrivial solution of Ax = 0, then every entry of x is nonzero.
15). The equation Ax = b is homogeneous if the zero vector is a solution.
16). If v 1 , · · · v 4 are in R4 and v3 = 2v 1 + v2 , then {v 1 , · · · , v4 } is linearly dependent.
17). If v 1 and v 2 are in R4 and v 2 is not a scalar multiple of v 1 , then {v 1 , v 2 } is linearly independent.
18). If v 1 , · · · , v5 are in R5 and v 3 = 0, then {v 1 , · · · , v 5 } is linearly dependent.
19). If v 1 , · · · v 4 are in R4 and {v1 , v2 , v3 } is linearly dependent, then {v1 , v2 , v 3 , v 4 } is also linearly
dependent.
20). An indexed set of two or more vectors {v 1 , v 2 , · · · , v p } is linearly dependent if and only if every vector in the set is a linear combination of the rest of the vectors.
21). If A is a 3 → 5 matrix and T is the transformation defined by T (x) = Ax, then the domain of
T is R 3 .
22). Every linear transformation is a matrix transformation.
23). Every matrix transformation is a linear transformation.
24). If A is an m → n matrix, then the range of the transformation x ↑ ↓ Ax is Rm .
25). A linear transformation T : Rn ↓ Rm is completely determined by its effect on the columns
of the n → n identity matrix.
26). If A is a 3 → 2 matrix, then the transformation x ↓ Ax cannot be one-to-one.
27). If A is a 3 → 2 matrix, then the transformation x ↓ Ax cannot be onto.
28). A linear transformation T : Rn ↓ R m is one-to-one if each vector in Rn is mapped to a
unique vector in Rm .
29). A linear transformation T : R n ↓ R m is onto if its standard matrix has a pivot position in
each row.
30). If the columns of A span Rn , then the columns are linearly independent.
31). Let A be an n → n matrix. If the equation Ax = 0 has a nontrivial solution, then A has fewer
than n pivot positions.
32). Let A be an n → n matrix. If there is a b in Rn such that the equation Ax = b is inconsistent,
then the transformation x ↑ ↓ Ax is not one-to-one.


F