Question

Let a quadratic form have the expression x2+y2+2z2+2xy+3xz with respect to the
standard basis B1 = f(1;0;0); (0;1;0); (0;0;1)g. Find its expression with respect to the
basis B2 = f(1;1;1); (0;1;0); (0;1;1)g

Answer 1

Answer on Question#40793 - Math - Linear Algebra

Task:

Let a quadratic form have the expression $x2 + y2 + 2z2 + 2xy + 3xz$ with respect to the standard basis $B1 = f(1;0;0)$ ; $(0;1;0)$ ; $(0;0;1)g$ . Find its expression with respect to the basis $B2 = f(1;1;1)$ ; $(0;1;0)$ ; $(0;1;1)g$

Solution:

Denote this quadratic form by $Q = x^{2} + y^{2} + 2z^{2} + 2xy + 3xz$ ;

The matrix of this quadratic form with respect to the

standard basis $\mathsf{B}1 = \{\mathsf{f}(1;0;0);(0;1;0);(0;0;1)\}$ is $\mathsf{A} = \begin{pmatrix} 1 & 1 & 3/2 \\ 1 & 1 & 0 \\ 3/2 & 0 & 2 \end{pmatrix}$ ;

This means that $A$ is a symmetric $n \times n$ matrix such that $Q = x^{T}Ax$ ;

$\mathbf{C} = \left( \begin{array}{lll}1 & 0 & 0\\ 1 & 1 & 1\\ 1 & 0 & 1 \end{array} \right)$ - is a transformation matrix from B1 to B2;

A1 = C^T A C - matrix of this quadratic form with respect to the basis B2;

Let find A1; A1 = $\begin{pmatrix} 1 & 1 & 1 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \end{pmatrix}$ , $\begin{pmatrix} 1 & 1 & \frac{3}{2} \\ 1 & 1 & 0 \\ \frac{3}{2} & 0 & 2 \end{pmatrix}$ , $\begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 1 \\ 1 & 0 & 1 \end{pmatrix} = \begin{pmatrix} \frac{7}{2} & 2 & \frac{7}{2} \\ 1 & 1 & 0 \\ \frac{5}{2} & 1 & 1 \end{pmatrix}$ , $\begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 1 \\ 1 & 0 & 1 \end{pmatrix} =$

$$\left( \begin{array}{c c c} 9 & 2 & 1 1 / 2 \\ 2 & 1 & 1 \\ 1 1 / 2 & 1 & 3 \end{array} \right);$$

So the expression of $Q$ with respect to the

basis $\mathrm{B}2 = \{(1;1;1);(0;1;0);(0;1;1)\}$ is $\mathrm{Q}1 = x^{T}A1x = (xyz)\left( \begin{array}{ccc}9 & 2 & \frac{11}{2}\\ 2 & 1 & 1\\ \frac{11}{2} & 1 & 3 \end{array} \right)\left( \begin{array}{c}x\\ y\\ z \end{array} \right) =$

$$9 x ^ {2} + y ^ {2} + 3 z ^ {2} + 4 x y + 1 1 x z + 2 y z;$$

Answer:

The expression of $Q$ with respect to the

basis $\mathrm{B}2 = \{\mathrm{f}(1;1;1);(0;1;0);(0;1;1)\}$ is $\mathrm{Q}1 = 9x^{2} + y^{2} + 3z^{2} + 4xy + 11xz + 2yz$