Answer to Question #86509 in Linear Algebra for RAKESH DEY

Question

Let T: P2 be defined by
T(a+bx+cx^2)= b+2cx+(a-b)*x^2.
Check that T is a linear transformation. Find the matrix of the transformation with respect to the ordered bases B1={x^2, x^2+x, x^2+x+1} and B2= {1,x,x^2} . Find the kernel of T.

Answer 1

Answer on Question #86509 – Math – Linear Algebra

Let $T : P_2$ be defined by

$$T \left( a + bx + cx^2 \right) = b + 2cx + (a - b) x^2 \, .$$

Check that T is a linear transformation. Find the matrix of the transformation with respect to the ordered bases $B_1 = \left\{ x^2 , x^2 + x , x^2 + x + 1 \right\}$ and $B_2 = \left\{ 1 , x , x^2 \right\}$. Find the kernel of T.

Solution:

Let $X_1 = a_1 + b_1 x + c_1 x^2$ and $X_2 = a_2 + b_2 x + c_2 x^2$ be two polynomials, and let $\alpha$ be a number. Then $X_1 + X_2 = a_1 + a_2 + \left( b_1 + b_2 \right) x + \left( c_1 + c_2 \right) x^2$,

$$T \left( X_1 + X_2 \right) = b_1 + b_2 + 2 \left( c_1 + c_2 \right) x \newline + \left( a_1 + a_2 - b_1 - b_2 \right) x^2 = T \left( X_1 \right) + T \left( X_2 \right)$$

and

$$T \left( \alpha X_1 \right) = \alpha b_1 + 2 \alpha c_1 x + \left( \alpha a_1 - \alpha b_1 \right) x^2 = \alpha T \left( X_1 \right) \, .$$

Hence, T is linear. Denoting the elements of the bases as $B_1 = \{ e_1 , e_2 , e_3 \}$ and $B_2 = \{ f_1 , f_2 , f_3 \}$, we have

$$T \left( e_1 \right) = 2 f_2 \, , \quad T \left( e_2 \right) = 2 f_2 - f_3 \, , \quad T \left( e_3 \right) = f_1 + 2 f_2 \, .$$

The matrix of this transformation, canonically defined as $T \left( e_i \right) = \sum_j T_{ji} f_j$, is then

$$\begin{pmatrix} 0 & 0 & 1 \\ 2 & 2 & 2 \\ 0 & -1 & 0 \end{pmatrix} \, .$$

This matrix is non-degenerate (its determinant is equal to -2), so that the kernel of T is the subspace formed by the null polynomial: ker T = {0}.