Determine whether the following vectors are linearly indepen

Determine whether the following vectors are linearly independent R^2: (a) (1 2), (1 3) yes no (b) (3 2), (-6 -4) yes no (c) (2 3), (4 7), (1 1) yes no (d) (1 1), (1 0) yes no

Solution

a)

Yes.

For two vectors to be linearly dependent they must be multiples of each other. These two vectors are clearly not multiples of each other so they are linearly independent vectors

b)

Second vector is obtained by multiplying first vector by -2

Hence the vectors are linearly dependent

c)

Three vectors all in R2. Any set of linearly independent vectors in R2 can have atmost 2 vectors

So given set is linearly dependent

d)

Yes it is linearly independent as clearly they are not multiples of each other.