Determine if the vector v x2 2x 1 is in the subspace of P

Determine if the vector v = x^2 + 2x - 1 is in the subspace of P^3 given by span{x^3 + x - 2, x^2 + 2x + 1, x^3 - x^2 + x)

Solution

In order to determine whether the vector v = x+2x-1 is in the given subspace of P₃, say V, we have to ascertain whether v is a linear combination of the basis vectors in V. Let A be the matrix with the coefficients of the basis vectors, and the coefficients of the vector v as columns, i.e. A =

1

0

1

0

0

1

-1

1

1

2

1

2

-2

1

0

-1

We will reduce A to its RREF as under:

Add -1 times the 1st row to the 3rd row; Add 2 times the 1st row to the 4th row

Add -2 times the 2nd row to the 3rd row;Add -1 times the 2nd row to the 4th row

Multiply the 3rd row by ½;Add -3 times the 3rd row to the 4th row

Multiply the 4th row by -1/2;Add -1 times the 4th row to the 2nd row    

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 I₄. This implies that the vector v and the basis vectors are linearly insdependent. Hence v is not in V.

1	0	1	0
0	1	-1	1
1	2	1	2
-2	1	0	-1