If A = [ 1 − 1 2 − 2 1 − 1 1 2...

Izabella Ponce

Izabella Ponce

Answered

2022-06-26

If
A = [ 1 1 2 2 1 1 1 2 3 ]
is the matrix representation of a linear transformation
T : P 2 ( x ) P 2 ( x )
with respect to the bases { 1 x , x ( 1 x ) , x ( 1 + x ) } and { 1 , 1 + x , 1 + x 2 } then find T. What is the procedure to solve it?

Answer & Explanation

crociandomh

crociandomh

Expert

2022-06-27Added 19 answers

Recall the the matrix of a transformation has as its columns the images of the domain basis vectors expressed relative to a basis of the codomain. Assuming that the first of the given bases is for the domain, this means that
T [ 1 x ] = 1 ( 1 ) 2 ( 1 + x ) + 1 ( 1 + x 2 ) = x 2 2 x .
The other two columns give you T [ x ( 1 x ) ] and T [ x ( 1 + x ) ], respectively. Now, express the general second-degree polynomial a + b x + c x 2 as a linear combination of the domain basis polynomials, i.e., as α ( 1 x ) + β x ( 1 x ) + γ x ( 1 + x ) and use linearity of T:
T [ a + b x + c y ] = α T [ 1 x ] + β T [ x ( 1 x ) ] + γ T [ x ( 1 + x ) ] . T [ a + b x + c y ] = α T [ 1 x ] + β T [ x ( 1 x ) ] + γ T [ x ( 1 + x ) ] .
Eden Solomon

Eden Solomon

Expert

2022-06-28Added 7 answers

The columns of the matrix of a transformation are the images of the basis vectors of the domain expressed relative to the basis of the codomain, i.e., it specifies a linear combination of the codomain basis vectors. So, the first column of A tells us that T [ 1 x ] = 1 ( 1 ) 2 ( 1 + x ) + 1 ( 1 + x 2 ) = x 2 2 x, and so on. From that, you should be able to work out what T does to the general polynomial a + b x + c x 2 . Alternatively, you might convert A to the standard basis and read the solution from that.

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