Each of the following questions is TRUE/FALSE. In the exam you must be able to j

Question

Each of the following questions is TRUE/FALSE. In the exam you must be able to justify your answers. The questions refer to the vectors u,v.w,x,y and z in the following regular hexagram. Note that all sides that look parallel are indeed parallel and have the same length. The vectors v and x are linearly independent The vectors z and -x-w+v are linearly dependent. Choose... The vectors w and z are linearly independent. The vectors u,v,w.x,y and z are linearly independent. Choose... The vector x is linearly dependent. The vectors x and v+w-x are linearly independent. Choose. The vectors x+w and z+u are linearly dependent. Choose The vectors v,x and z are linearly independent. Choose. Choose. Choose Choose

Explanation / Answer

1. The vectors u and x are linearly dependent. True, as u -x = 0.

2. The vectors z and -x-w+v are linearly dependent. True,as the vector -x-w+v is a scalar multiple of z.

3. The vectors w and z are linearly dependent. True, as w-z = 0.

4. The vectors u,v,w,x., y,z are linearly dependent. True,as (u-x)+(v-y)+(w-z) = 0+0+0 = 0.

5. The vector x is linearly dependent. False, as x is non-zero.

6. The vectors x and v+w-x are linearly dependent. False as no linear combination of x and v+w-x can equal 0.

7. The vectors x+w and z+u are linearly dependent. True, as (x+w)-(z+u) = 0.

8. The vectors v,x and are linearly dependent. False as no linear combination of v,x and z can be zero.

                

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