Answer to Question #154538 in Linear Algebra for Ashweta Padhan

Question #154538

Is the quadratic form x12+x22+2x1x3+4x1x3+3x32 quadratic? Justify your answer


1
Expert's answer
2021-01-11T17:35:05-0500

A quadratic form is a polynomial with terms all of degree two. 

Quadratic forms are homogeneous quadratic polynomials in nn variables.

In our case n=3,n=3, and we have ternary quadratic form

f(x1,x2,x3)=1x12+0x1x2+6x1x3+1x22f(x_1, x_2, x_3)=1\cdot x_1^2+0\cdot x_1x_2+6\cdot x_1x_3+1\cdot x_2^2

+0x2x3+1x32+0\cdot x_2x_3+1\cdot x_3^2

The quadratic form f(x1,x2,x3)f(x_1, x_2, x_3) is determined by the matrix

A=[103010301]A=\begin{bmatrix} 1 & 0 & 3 \\ 0 & 1 & 0 \\ 3 & 0 & 1 \end{bmatrix}


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