Linear Algebra: UNIT I: Vector Spaces
Linear span: Example Solved Problems
Important Example Solved Problems - Engineering Maths or Mathematics - Linear span: Example Solved Problems
Linear span
Example Problems
Example 12
Verify 2x3−
2x2[P (x)]+12x – 6 is a linear combination of x3− 2x2[Q(x)]−5x−3
and 3x3−5x2[R (x)]−4x−9 in P3 (R).
Solution:
Let
v = av1+bv2
2x3 − 2x2 + 12x − 6 = a (x3
− 2x2 − 5x − 3) + b (3x3 − 5x2 − 4x−9)
2x3−
2x2+12x−6=x3 (a + 3b) + x2 (−2a − 5b) + x (−5a−4b)
+ (− 3a − 9b)
Equating
the co−efficients on both sides we get
a+3b=2 ...(1)
−2a−5b=−2 ...(2)
−5a−4b
= 12 ...(3)
−3a−9b=−6 ...(4)
From
(1) and (2); −2a−5b=−2
2a+6b
= 4
b = 2.
From
(1); a +3 (2) = 2
a=2−6
a=−4.
Equation
− 5a−4b= 12 holds good.
−5(−4)−4
(2) = 12
20−8=
12
Equation−3a−9b−6
is also satisfied.
−3(−4)
− 9(2)=−6
12−18=−6
Hence
2x3− 2x2 + 12x−6 is a linear combination of x3−2x2−5x−3
and 3x3−5x−4x−9.
P(x) is
a linear combination of Q (x) and R (x).
Example 13
Verify 3x3 –
2x2 P(x) +7x+8 is a linear combination of x3−2x2 Q(x) −5x−3
and 3x3−5x2 R (x)−4x – 9 in P3 (R).
Solution:
Let
v=av1+bv2
3x3
−2x2+7x+8 = a (x3 − 2x2 − 5x − 3) + b (3x3
– 5x2 − 4x − 9)
3x3− 2x2+7x+8 = x3(a
+ 3b) + x2(− 2a − 5b) + x(− 5a − 4b) + (− 3a − 9b)
Equating
the coefficients on both sides, we get
a+3b=3
...(1)
2a−5b=−2 ...(2)
−5a−4b=7 ...(3)
−3a−9b=8 ...(4)
From
(2) and (1);
−
2a − 5b = − 2
2a+6b=6
:.
b = 4.
From
(1):
a+12=3
a=−9.
From
(3): −5(−9)−4(4)=7
45−16
≠ 7 and also
From
(4): −3(−9)−9 (4) ≠ 8
These
two conditions are not satisfied.
Hence
3x2−2x2+7x+8 is not a linear combinations x3−2x2−5x−3
and 3x3−5x2−4x−9.
P(x) is not a linear combination of Q (x) and
R (x).
Example 14
Let S = { (1, 2), (2,
1) }; V=R2. Prove that it is a linear span.
Solution:
Let
us consider (x, y) ∈
V
(x, y) = α(1, 2) + β(2, 1) = (α, 2α)+(2β, β)
(x,
y) = (α + 2β, 2α+β)
α + 2β = x ... (1)
2α
+ β = y
From
equation (2);
β = 2x−y
2α = y – ( (2x– y) / 3 )
2α
= [ 3y – 2x + y ] / 3
2α
= (4y−2x) / 3
α
= (2y−x) / 3
(x,
y) = ( 2y−x / 3 )(2, 1) + ( 2x−y / 3 )(2, 1)
(x, y) is a linear combination of S
(x, y) ∈ L (S)
V
L(S)
and L (S)V
⇒ L(S) = V.
Linear Algebra: UNIT I: Vector Spaces : Tag: maths, mathematics : - Linear span: Example Solved Problems
Linear Algebra: UNIT I: Vector Spaces
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