Vectors and Linear Independance

[bas_02]

Not sure if im correct. My options are "ALWAYS", "NEVER" and "SOMETIMES".

 

[Deleted member 4993]

Not sure if im correct. My options are "ALWAYS", "NEVER" and "SOMETIMES".

View attachment 20152

How did you come to those conclusions?

What is the definition of linear independence of vectors?

What does spanning mean in case of vectors?

 

[HallsofIvy]

Since \(\displaystyle v= v_1+ v_2+ v_3\) and \(\displaystyle v=v_1+ 4v_2- 5v_3\) it follows that \(\displaystyle v_1+ v_2+ v_3= v_1+ 4v_2- 5v_3\) so subtracting \(\displaystyle v_1+ v_2+ v_3\) from both sides, \(\displaystyle 0= 3v_2- 4v_3\). Now what does that tell you?

If you have "n" independent vectors in an n-dimensional space they must span the space.

What is the definition of "basis" for a vector space?

 

[bas_02]

Since \(\displaystyle v= v_1+ v_2+ v_3\) and \(\displaystyle v=v_1+ 4v_2- 5v_3\) it follows that \(\displaystyle v_1+ v_2+ v_3= v_1+ 4v_2- 5v_3\) so subtracting \(\displaystyle v_1+ v_2+ v_3\) from both sides, \(\displaystyle 0= 3v_2- 4v_3\). Now what does that tell you?

If you have "n" independent vectors in an n-dimensional space they must span the space.

What is the definition of "basis" for a vector space?

Thanks. I got my answer now

 

[Singleton]

Your initial answers were wrong. What are your new answers? By the way I think it's – 6v_3 above.

 

[bas_02]

Your initial answers were wrong. What are your new answers? By the way I think it's – 6v_3 above.

My new answer is never, sometimes and always

 

[Steven G]

Now can we see your work?

 

[Singleton]

My new answer is never, sometimes and always

Sounds good to me.

 

[Singleton]

Your initial answers were wrong. What are your new answers? By the way I think it's – 6v_3 above.

Oh wait a minute, need to take another look at 3.

 

[bas_02]

Now can we see your work?

My reasoning is that if for 1) v2 depends on v3, they are never linearly independent
for 2) They sometimes span R^3. We don't have enough info to conclude whether they do or don't. It could also be that all vectors map to a 1d line or 2d plane. We don't know.

For 3) if v, v1 and v2 are basis vectors, this implies they r all linearly independent and span r3. This must also mean that v3 also spans r3 nd is linearly independent.

 

[Singleton]

Hi bas_02 in the last one, v_1 went missing. Is that ok?

 

[Steven G]

Part B) V can be expressed in terms of V1, V2 and V3 so V contributes nothing to the span of {V1, V2, V3}. Now V2 can be expressed in terms of V3. What does that tell you?