Linear Combination


Note:
i. Vector Space under consideration:( R2 +, .) over R
ii. Chosen elements in R2 are represented by " Red " dots in the diagrams below.

Single element

α . (x1 , x2) = (α . x1 , α . x2)

Case 1 : x ≡ (1 , 2)




Case 2 : x ≡ (0 , 0)


---- Try it yourself for other values of α ----

Two elements

α . (x1 , x2) + β . (y1 , y2) = (α . x1 + β . y1 , α . x2 + β . y2)

Case 1 : x ≡ (1 , 2) , y ≡ (2 , 4)




Case 2 : x ≡ (1 , 2) , y ≡ (2 , 5)


---- Try it yourself for other values of α and β ----

More than two elements

α . (x1 , x2) + β . (y1 , y2) + γ . (z1 , z2) =
(α . x1 + β . y1 + γ . z1 , α . x2 + β . y2 + γ . z2)

Case 1 : x ≡ (1 , 2) , y ≡ (2 , 4), z ≡ (3 , 5)




Case 2 : x ≡ (1 , 2) , y ≡ (5 , 3), z ≡ (6 , 2)


---- Try it yourself for other values of α , β and γ ----