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Homework Statement
Indicate whether the following is true or false. Explain your answer.
If [tex]\overline{u}, \overline{v}, \overline{w}[/tex] are vectors in [tex]R^{n}[/tex] such that {[tex]\overline{u}, \overline{v}[/tex]} and {[tex]\overline{v}, \overline{w}[/tex]} are each linearly independent sets, then {[tex]\overline{u}, \overline{v}, \overline{w}[/tex]} is a linearly independent set.
The attempt at a solution
I think that the above is false because for {[tex]\overline{u}, \overline{v}[/tex]} and {[tex]\overline{v}, \overline{w}[/tex]} to each be linearly independent sets, they must have two entries for each vector, as this would give them trivial solutions only. Therefore, they would each be a 2 x 2 matrix. However, {[tex]\overline{u}, \overline{v}, \overline{w}[/tex]} is not a linearly independent set because it would form a 3 x 2 matrix. This would automatically have a free variable, and so infinite solutions would result.
Indicate whether the following is true or false. Explain your answer.
If [tex]\overline{u}, \overline{v}, \overline{w}[/tex] are vectors in [tex]R^{n}[/tex] such that {[tex]\overline{u}, \overline{v}[/tex]} and {[tex]\overline{v}, \overline{w}[/tex]} are each linearly independent sets, then {[tex]\overline{u}, \overline{v}, \overline{w}[/tex]} is a linearly independent set.
The attempt at a solution
I think that the above is false because for {[tex]\overline{u}, \overline{v}[/tex]} and {[tex]\overline{v}, \overline{w}[/tex]} to each be linearly independent sets, they must have two entries for each vector, as this would give them trivial solutions only. Therefore, they would each be a 2 x 2 matrix. However, {[tex]\overline{u}, \overline{v}, \overline{w}[/tex]} is not a linearly independent set because it would form a 3 x 2 matrix. This would automatically have a free variable, and so infinite solutions would result.