Dse 2008 ques 16

[cheesecake]

Consider the following two functions mapping points on the plane back to points on the plane
f (x1, x2) = (x1 +1, x2 + 1)
g (x1, x2) = (x2, x1)
Which of the above is a linear function?

[Sris]

hey...just watch the linear transformation video of khan academy...they have explained it beautifully...

[Aashna]

only second function is linear.

For function 1  - if you take (x1 +1,x2+1) and (x3+1,x4+1)
If you add them they should lie in the set - (one of the properties to check linear)
but by adding we get - (x1+x3+2,x2+x4+2) - The part +2 does not lie in our set. We can only have +1 there.
Hence not linear

[cheesecake]

Got it. Thanks!
Also please help me out with ques 60.
Suppose v1, v2, v3 are three vectors in 3-dimensional space and are linearly independent. Then the vectors v1+v2, v2+v3, v1+v3 are?

The solution says they are linearly dependent but i'm getting linearly independent. I am attaching my answer.

Because v1, v2, v3 are linearly independent,
xv1 + yv2 + zv3 = 0 implies that x=y=z=0
Now if v1+v2, v2+v3, v1+v3 are linearly independent,
a(v1+v2) + b(v2+v3) + c(v1+v3) = 0 would imply a=b=c=0
Opening the brackets we get the following three equations
a+c =0
a+b =0
b+c= 0
This implies that a=b=c=0
Is this not correct?

[Sris]

Hey v1 v2 and v3 are linearly dependent...not independent...

Hey from where to read this concept...i think i havenot understood the concept

[cheesecake]

Oh god i can't believe i read the question wrong! Thanks for pointing it out sris..
I did it from manmohan singh bhasin. I don't know if you took linear algebra and calculus as an optional. This was the prescribed reading. If not then do it from chiang.