Linear Combinations

Página 1

Linear combination:

e.g. a = (3, 4) and b = (6,3)

Linear combination would be an upscale of the vector e.g.

7a + 4b = (21, 28) + (24, 12) = (45, 40)

Span of the vectors a & b = (a, b) = R2

e.g. Span(a) = c*a ... 4a .... 3a ..... 2a ..... 17a ..... 12a

Vectors i and j

i = (1, 0) & j = (0, 1)

Summary:

Span of vectors represent any combination within R2 or R3 or R4 depending on the vector size

To find any location of a vector use of the formula: ....

Página 2

Linear Independence

Linearly dependent set of vectors would be e.g. a + b = c

(2, 3) + (9, 10) = (11, 13) <---- the vector behind the equal sign is linearly dependent on a + b

Span of {(2,3), (4,6)} = c1 + 2c2 = c3 * (2,3)

Span of v1 & v2 = R2 > the vector is in the 2 dimensional space

Página 3

Definition of linear Dependence/Independence

Linearly independent

Linearly dependent

Span

R5 = a(1, 2, 3, 4, 5) + b(6,7,8,9,10)

R4 = a(1, 2, 3, 4) + b(5, 6, 7, 8)

R3 = a(1, 2, 3) + b(4, 5, 6)

R2 = a(1, 2) + b(3, 4)