Question

2. Determine whether the first vector can be expressed as a linear combination
of the other two 4x3 + 2x2 – 6 is a linear combination of x3 - 2x2 + 4x + 1
and 3x3
6x2 + x + 4.
-​

Answer 1

Given: three polynomials

• first one = 4x³ + 2x² - 6
• second one = x³ - 2x² + 4x + 1
• third one = 3x³ - 6x² + x + 4

To find: whether the first one is a linear combination of the other two

Solution:

We consider,

• 4x³ + 2x² - 6 = a (x³ - 2x² + 4x + 1) + b (3x³ - 6x² + x + 4), where a and b are scalars

Equating from both sides, we get

• a + b = 4 .....(1)
• - 2a - 6b = 2 or, a + 3b = - 1 .....(2)
• 4a + b = 0 .....(3)
• a + 4b = - 6 .....(4)

From (1) and (2), eliminating a, we get

• 2b = - 5 or, b = - 5/2

Putting b = - 5/2 in (1), we get a = 13/2

• Now we put a = 13/2, b = - 5/2 in left hand side of (3)
• = 26 - 5/2 = 47/2 ≠ 0

• Again we put a = 13/2, b = - 5/2 in left hand side of (4)
• = 13/2 - 10 = - 7/2 ≠ - 6

So we cannot find satisfactory values of a and b.

Answer: we can not write the first term as a linear combination of the other two.