D Section 4.6 Change of Basis - Abdulla Eid

Section 4.6

id Change of Basis ulla E Dr. Abdulla Eid bd College of Science Dr. A MATHS 211: Linear Algebra

Dr. Abdulla Eid (University of Bahrain)

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id Goal: E 1 Relation between bases. Dr. Abdulla 2 Transition matrix.

Dr. Abdulla Eid (University of Bahrain)

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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.

Dr. Abdulla Eid (University of Bahrain)

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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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