D Section 4.6 Change of Basis - Abdulla Eid
[Pages:9]Section 4.6
id Change of Basis ulla E Dr. Abdulla Eid bd College of Science Dr. A MATHS 211: Linear Algebra
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id Goal: E 1 Relation between bases. Dr. Abdulla 2 Transition matrix.
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Relation between bases
Example 1
Let V = R2. and let B =
id be two bases for V .
1 1
,
1 -1
and B =
lla E find
du 2
Ab 1
2
.
B
. 2
r 2
.
D 2 B
2 -1
,
-1 1
3 Find v , w such that (v )B = (5, 6) and (w )B = (5, 6)
4 Find v such that (v )B = (-2, 2) and then find (v )B .
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Transition Matrix
pause
Theorem 2 Let V be a vector space with two bases B (old basis) and B (new basis).
id There is a matrix P such that for any v in V we have lla E (v)B = P ? (v)B
Abdu We will often denote P by PBB (transition matrix). Dr. Question: How to find the transition matrix? Method 1:
[ new basis | old basis ] [In| transition matrix from old to new]
Method 2: The column of the transition matrix are the coordinate vectors of the old basis relative to the new basis.
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Example 3
Let V = R2. and let B =
1 1
,
1 -1
and B =
be two bases for V .
Eid find lla 1 Find the transition matrix from B to B . Dr. Abdu 2 Given (v)B = (-1, -3). Find (v)B ?
2 -1
,
-1 1
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Example 4
Let V = P2. and let B = 1, X , X 2 and B = 1, 1 + X , 1 + X + X 2 be two bases for V .
id find E 1 Find the transition matrix from B to B . Dr. Abdulla 2 Given p(X ) = 4 - 2X + 6x2. Find (p(X ))B ?
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Example 5
Let V = P2. and let B = 1, X , X 2 and B = 1, 1 + X , 1 + X + X 2 be two bases for V .
id find E 1 Find the transition matrix from B to B . Dr. Abdulla 2 Given p(X ) = 4 - 2X + 6x2. Find (p(X ))B ?
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Inverse of a transition matrix
Question: What is the inverse of a transition matrix? What is PB-1 B ?
Example 6
Let V = R2. and let B =
Eid two bases for V .
2 2
,
1 -2
and B =
ulla find bd 1 Find the transition matrix from B to B . A 2 Find the transition matrix from B to B. Dr. 3 Multiple the two matrices together.
-1 2
,
3 0
be
Theorem 7
PB B = PB-1 B
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