Linear Independence: V1-V4 in R4 - True or False?

[thrive]

Homework Statement

If V1...V4 are linearly independent vectors in R4, then {V1, V2, V3} are also linearly independent. True or False.

The Attempt at a Solution

My solution involved reducing the problem down the 3 vectors in R3. Then show a counter example of this in R3 although I have not been able to come up with one.

 

[gabbagabbahey]

Why not assume that $\{V_1,V_2,V_3\}$ are linearly dependent, what would that mean?

 

[thrive]

it would mean that the answer is false...

 

[gabbagabbahey]

Obviously, but that doesn't prove anything.

Start by assuming $\{V_1,V_2,V_3\}$ are linearly dependent, under that assumption, what could you say about the equation $c_1V_1+c_2V_2+c_3V_3=0$?

What would that imply about the equation $c_1V_1+c_2V_2+c_3V_3+c_4V_4=0$?

 

[thrive]

it would mean there is some linear combination of c1, c2, c3 (not all zero) that would solve that first equation. In the second equation it would imply that not necessarily would there be a c4 to cancel the V4 term?

 

[gabbagabbahey]

it would mean there is some linear combination of c1, c2, c3 (not all zero) that would solve that first equation.

Right. So choose those values of c1, c2, and c3 and plug them into the second equation.

In the second equation it would imply that not necessarily would there be a c4 to cancel the V4 term?

If $c_1V_1+c_2V_2+c_3V_3=0$ then, $c_4=0$ would surely satisfy the equation $c_1V_1+c_2V_2+c_3V_3+c_4V_4=0$ right?

But if $\{V_1,V_2,V_3\}$ are linearly dependent, then there would exist $c_1$, $c_2$ and $c_3$ not all equal to zero, so even if $c_4=0$ there would exist $c_1$, $c_2$, $c_3$ and $c_4$ not all equal to zero that would make $c_1V_1+c_2V_2+c_3V_3+c_4V_4=0$, which would mean what?

 

[thrive]

which would mean that the answer to the question is false

 

[gabbagabbahey]

No, if there exist $c_1$, $c_2$, $c_3$ and $c_4$ not all equal to zero that would make $c_1V_1+c_2V_2+c_3V_3+c_4V_4=0$, would the set $\{V_1,V_2,V_3,V_4\}$ be linearly independent or dependent?