If B is the standard basis of the space Pg of polynomials, then let B- (1.2). Us
ID: 3116561 • Letter: I
Question
If B is the standard basis of the space Pg of polynomials, then let B- (1.2). Use coordinate vectors to test the linear independence of the set of polynomials below Explain your work. 1-8-8 Write the coordinate vector for the polynomial 1-81-t Wnite the coordinate vector for the polynomial t+6r Write the coordinate vector for the polynomial 1 +t-8 each coordinate vector a column of the matrix. If possible, write the matrix in reduced echelon form. 1 0 1 0 1 1 -16 0 Click to select your answerls). social eExplanation / Answer
The coordinate vectors of v1 = 1-8t2-t3, v2 = t+6t3, v3 =1+t-8t2 with respect to the standard basis B = {1,t,t2,t3} of P3 are (1,0,-8,-1)
The coordinate vectors of v1 = 1-8t2-t3 with respect to the standard basis B = {1,t,t2,t3} of P3 is (1,0,-8,-1)T.
The coordinate vectors of v2 = t+6t3 with respect to the standard basis B is (0,1,0,6)T.
The coordinate vectors of v3 =1+t-8t2 with respect to the standard basis B is (1,1,-8,0)T.
Let A be the 4x3 matrix with v1,v2,v3 as columns. Then A =
1
0
1
0
1
1
-8
0
-8
-1
6
0
The matrix A can be reduced to its RREF as under:
Add 8 times the 1st row to the 3rd row
Add 1 times the 1st row to the 4th row
Add -6 times the 2nd row to the 4th row
Interchange the 3rd row and the 4th row
Multiply the 3rd row by -1/5
Add -1 times the 3rd row to the 2nd row
Add -1 times the 3rd row to the 1st row
Then the RREF of A is
1
0
0
0
0
1
0
0
0
1
0
0
Now, apparently, the columns of the RREF of A, and hence the columns of A, i.e. the coordinate vectors of v1,v2,v3, with respect to the standard basis B = {1,t,t2,t3} of P3, are linearly independent.
1
0
1
0
1
1
-8
0
-8
-1
6
0