MHB Solve Vector Space Question: Get the Solution Now

[LearnerJr]

View attachment 6276

How do you solve this question I just need a solution

 

[Greg]

Hello LearnerJr and welcome to MHB! :D

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Can you post what you have done so far?

 

[LearnerJr]

Hello LearnerJr and welcome to MHB! :D

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Can you post what you have done so far?

Yes of course. For 2a I know a bit of indirect proof and assume the result
Is not true. But not sure still how to perceive this question in the long run.
2b I know if it's a basis it's vectors have to be linearly independent, they span V.but I still can't solve it.

 

[HallsofIvy]

Consider the equation $af(v_1)+ bf(v_2)+ cf(v_3)= 0$. In order to show that $f(v_1)$, $f(v_2)$, and $f(v_3)$ are independent we must show that a= b= c= 0.

Since f is a linear transformation, $af(v_1)+ bf(v_2)+ cf(v_3)= f(av_1+ bv_2+ cv_3)= 0$. Since the kernel of f is only the 0 vector, we must have $av_1+ bv_2+ cv_3= 0$. But we were given that $v_1$, $v_2$, and $v_3$ are independent so a= b= c= 0 as we wished.