1.
prove that any set consisting of two non-zero vectors is linearly independent if, and only if, one
vector is not a scalar multiple of the other.
Solution
Let's we have vectors x_1, x_2, ..., x_n then these n vectors are linearly dependent if we can
write it like this: a_1x_1 + a_2x_2 + ... + a_nx_n = 0 Where a_1, a_2, ..., a_n are not all zero.
Ex: Lets verify that (1, 2) and (2, 4) are linearly dependent: a(1, 2) + b(2, 4) = 0 If we can find a
and b such that the above holds then they are linearly dependent. Notice that if we let a = -2 and
b = 1, the above equation holds. Therefore, they are linearly dependent. Now what happens if the
only solutions to a_1x_1 + a_2x_2 + ... + a_nx_n = 0 for a_1, a_2,..., a_n are all equal to zero?
That would mean that they are linearly independent. Ex: Let's verify that (1, 0) and (0, 1) are
linearly independent: Let's see if we can find scalars such that a(1, 0) + b(0, 1) = 0 Adding the
two we get: (a, b) = 0 This shows that a = b = 0. Therefore, they are linearly independent. Now if
vectors are coplanar that means they lie on the same plane. To verify whether two vectors are
coplanar (lets call them b and c) you must first find a vector that is perpendicular to the plane
which we'll call a. Now if we have that a • b = 0 and a • c = 0; where "•" denotes the
dot product or the scalar product, or in higher level math the inner product. That means that b
and c both lie on the plane. Therefore, they are coplanar. If lets say we had that a • b = 0 but a
• c ? 0 That would mean that b is on the plane but c is not. Therefore, they are not coplanar.
Points are collinear if they lie on the same line. So if you had three points on a line lets call them
A, B, and C then they are collinear if the slopes of the lines AB and BC (which are the lines
joining each of those points) have the same slope. (1) That's becuase your only working with a
third dimension. If you were in 4-space then the statement would actually be false. (2) Yes. They
lie on the same plane, therefore, we'd be able to find a scalar such that ax_1 = x_2, where x_1
and x_2 are vectors and a is a scalar. (3) Yes which is pretty much the same reason as (2) only
that we have three vectors. The second part is also true. Hope this helps!