The coordinate vector of \displaystyle{\left[{\mathbf{{{p}}}}

nagasenaz 2020-12-17 Answered

The coordinate vector of [p (t) = 6 + 3t t2] relative to the basis {B} ={1 + t, 1 + t2, t + t2}

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Expert Answer

Nathaniel Kramer
Answered 2020-12-18 Author has 78 answers

Any arbitrary vector in P2 can be written as,
[p (t) = a ( 1 + t2 ) + b ( t + t2 ) + c ( 1 + 2 t + t2 )]
Thus, for the vector [p ( t )=6 + 3 t  t2] can be written as,
1) [a ( 1 + t ) + b ( 1 + t2 ) + c ( t + t2 )=6 + 3t  t2]
On comparing the terms of both the sides in the above equation (1) gives the following equations.
a + b=6
a + c=3
b + c=1
The representation of the linear system in a matrix is,
[110101011][acd]=[631]
The solution of the above system gives as by solving the augmented matrix
[110610130111]
Use Elementary row transformations to reduce the augmented matrix to row reduced Echelon form.
Step 1:
[110  6101  30111]R2  R2  R1[110   601   130111]
Step 2
[110   601   130111]R3  R3 + R2[110   601   130024]
Step 3:
[110   601   130024]R1  R1 + R2 12 R3[100   501   130024]
Step 4:

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