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lec9.ppt
1. NPTEL β Physics β Mathematical Physics - 1
Lecture 9
Linear independence
ο·Determine whether u and v are linearly independent
i) π’ = (1, 2), π£ = (3, β5), (ii) π’ = (1, β3), π£ = (β2π’)
2 vectors are said to be linearly dependent if one is multiple of another.
a)
b)
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u and v are independent
u and v are dependent for π£ = β2π’
ο· Determine whether 3 vectors
π’ = (1, 1, 2), π£ = (2, 3, 1) and π€ = (4, 5, 5) are linearly independent.
1 2 4 0
π₯ [1] + π¦ [3] + π§ [5] = [0]
2 1 5 0
If this set has a non-zero solution for (x, y, z) then they are linearly dependent.
The students should check this.
Change of basis
Let {π1, π2 β¦ β¦ β¦ ππ} is a basis of a vector space v and {π1 β¦ β¦ β¦ β¦ ππ} is another
basis. Suppose there is a relation that exists between the two bases such that,
π1 = π11π1 + π12π2 + β¦ β¦ β¦ π1πππ
π2 = π21π1 + π22π2 + β¦ β¦ β¦ π2πππ
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ππ = ππ1π1 + ππ2π2 + β¦ β¦ β¦ πππππ
Then the transpose, P of the above matrix of coefficients is called the basis
matrix.
Theorem 1
Let P be a basis matrix from a basis {e;} to a basis {ππ } and Q be the change of
basis matrix from the basis {ππ } to {ππ} back. Then P is invertible and π = πβ1
Proof ππ = βπ πππ π
π =1
π
(1)
(2)
Also ππ = βπ
π=1 ππ π π
π
Substituting (2) in (1)
ππ = βπ πππ(βπ πππππ ) = βπ (βπ π π ) π
π =1 π=1 π=1 π =1 π π π π π
Now, βπ ππππππ = πΏππ
2. NPTEL β Physics β Mathematical Physics - 1
Where πΏππ is the Kronecker delta function with the following properties,
πΏππ = 1 for π = π
= 0 for iβ π So, πΆππ = πΏππ so ππ = 1 Or π = πβ1 (proved)
Example
Consider the following bases in π 2.
π1 = {π’1 = (1, β2), π’2 = (3, β4)}
π2 = {π£1 = (1,3), π£2 = (3,8)}
(i) Find the components of an arbitrary vector (π) in π 2 in basis π1 = {π’1, π’2}.
(ii) Write the change of basis matrix P from π1to π2. To do this we have to write
π£1 and π£2 in terms of π’1 and π’2.
π
Solution
(π) = π₯ ( ) + π¦ ( ) β π₯ + 3π¦ = π and β2π₯ β 4π¦ = π
π 1
β2 β4
3
3 1
Thus, (π1π)π 1 = (β2π
β 2
π) π’1 + (π + 2
π)
π’2
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