Show that the set is linearly dependent by finding a nontrivial linear combination of vectors in the set whose sum is the zero vector. (Use s1, s2, and s3, respectively, for the vectors in the set.) S = {(3,4),(−1,1),(2,0)}

Let's define:

s1 = (3, 4)

s2 = (-1, 1)

s3 = (2, 0)

The system will be linearly dependent if we can write one of the vectors as a linear combination of the other two, so for example:

s1 = a*s2 + b*s3

where a and b are real numbers.

So let's try to solve this:

(3, 4) = a*(-1, 1) + b*(2, 0)

Note that the second term in the left only modifies the x-component. And the y-component in the left is 4, so we can conclude that a = 4 (because the first term is the only that has an y-component different than zero)

(3, 4) = 4*(-1, 1) + b*(2, 0)

(3, 4) = (-4, 4) + b*(2, 0)

Now we need to solve:

3 = -4 + b*2

3 + 4 = b*2

7 = b*2

7/2 = b

Then the linear combination is:

s1 = 4*s2 + (7/2)*s3

(3, 4) = 4*(-1, 1) + (7/2)*(2, 0)

Then the set is linearly dependent.