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 hav 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)

 

[Architeuthis Dux]

Hi there, it looks like you are working with linear combinations in your lecture notes. A linear combination is a mathematical operation that involves multiplying vectors by scalars (numbers) and adding them together. In this example, you are trying to show that the vectors w1, w2, and w3 can be written as a linear combination of the vectors v1 and v2. This means that you can find values for the scalars a1 and a2 that, when multiplied by v1 and v2 and added together, will give you the same values as w1, w2, and w3. In order to find these values, you can set up a system of equations by equating the components of w1, w2, and w3 to the linear combination of v1 and v2. By solving this system of equations, you can find the values of a1 and a2 that will make the linear combination work. I hope this helps clarify the concept of linear combinations for you. Let me know if you have any other questions!