The vectors P1, P2, P3, … are said to be linearly dependent if the real numbers k1, k2, k3, .. not all zero can be found so that
k1P1 + k2P2 + ….. + knPn = 0.
Since it is possible to solve for e.g. P1= -k2P2/k1-k3P3/k1-….
This means that all the vectors lie on the same line through the origin.
Conversely, if two vectors lie along the same line they are linearly dependent vectors.
A set of vectors which are not dependent are said to be
linearly independent.
An example of linearly independent vectors are E1 = (1,0) and E2 = (0,1).
They form the basis for the two dimensional vector-space. In favt any two linearly-independent vectors can form the basis of a two dimensional vector space. The dimension is identical to the number of vectors necessary.
Two geometric formulae:
Menelaos theorem: A line cuts the sides BC, CA and AB of a triangel in the points L, M and N respectively. If L = xB + x’C + y’A, M = yC + y’A and
N = zA + z’B, where x +x’ = y + y’ = z +z’ = 1,
then xyz = -x’y’z’.
