If the transformation is from R 3 → R is T { a , b...

veneciasp

veneciasp

Answered

2022-06-29

If the transformation is from R 3 R is
T { a , b , c } = 0 π 2 a e t + 2 b sin ( t ) + 3 c cos ( t ) d t
How to find the standard matrix?

Answer & Explanation

Alexia Hart

Alexia Hart

Expert

2022-06-30Added 19 answers

By "standard matrix" maybe you mean the matrix with respect to the basis
B = { ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) , ( 0 , 0 , 1 ) } for R 3
We could rewrite T this way:
T : R 3 R
T ( v ) = T ( ( x , y , z ) ) = 0 π 2 x e t + 2 y s i n ( t ) + 3 z c o s ( t ) d t
= 2 x 0 π e t d t + 2 y 0 π s i n ( t ) d t + 3 z 0 π c o s ( t ) d t
To write the matrix of a transformation with respect to some basis we should first apply the transformation on the basis vectors, so for example:
T ( ( 1 , 0 , 0 ) ) = 2 ( 1 ) 0 π e t d t + 2 ( 0 ) 0 π s i n ( t ) d t + 3 ( 0 ) 0 π c o s ( t ) d t
= 2 0 π e t d t
= 2 ( e π e 0 )
= 2 ( e π 1 )
To get the matrix you should apply T to the other base vectors and form the matrix with 3 rows and 1 column.
T ( ( 0 , 1 , 0 ) ) = 2 0 π s i n ( t ) d t = 4
T ( ( 0 , 0 , 1 ) ) = 3 0 π c o s ( t ) d t = 0
So the matrix of T with respect to B is: ( 2 ( e π 1 ) 4 0 )
To apply T to any ( x , y , z ) R 3 simply multiply:
T ( ( x , y , z ) ) = ( x y z ) ( 2 ( e π 1 ) 4 0 )
gaiaecologicaq2

gaiaecologicaq2

Expert

2022-07-01Added 6 answers

You find the matrix of a transformation by calculating what the transformation does to a basis of the domain of the transformation.

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