What Are Linear Combinations in Vector Mathematics?

[Chadlee88]

Linear combinations?? :S

Hey, could som1 please explain linear combinations. I copied down the lecture notes but I'm not understanding this example may have typo from the note takin

Example: Show that each of the vectors

w1 = (1, 0), w2= (0, 1) and w3 = (3, 3) are a linear combination of

v1 = (2,2) and v2 = (3,2)

write w1 = a1(2, 2) + a2(3, 2)

Equate components:

1. 2a1+3a2 = 1
2. 2a1+2a2 = 0

1. - 2.

a2 = 1, plug back into 2. a1 = -1

ie. w1 -v1+v2

Similarly w2 = 3v1/2 - v2
& w1 = 3v1/2 + 0v2

Tanx

 

[daveb]

What it means is that suppose you have two vectors v1 and v2 (as written above) Show that each of w1, w2 and w3 can be written as a1v1 + a2v2, where a1 and a2 are numbers (they will be different values for the solution to w2 than they are for w1).

For example, suppose the vecors v1 and v2 are respectively (1,0) and (0,1). Then the vector w1 = (3, -2) could be written as 3v1 - 2v2 = 3(1,0) -2(0,1) = (3,0) - (0,2) = (3,-2)