## Analytic expression for basis change

The quadratic form $$g$$ on the vector space $$\mathbb R^4$$ has w.r.t. the canonical basis $$K$$ analytic expression $$g(u)=2x^2+2xy-y^2-2yt-t^2$$, where $$u=(x, y, z, t)^T$$.

Find its analytic expression w.r.t. the basis

$$X=\{(1, 1, 1, 1)^T,(1, 1, 1, 0)^T,(1, 1, 0, 0)^T,(1, 0, 0, 0)^T\}$$.

Determine $$g(u)$$ for the vector $$u$$, which has w.r.t. the basis $$X$$ coordinates $$[u]_X=(3, 1, 0, 0)^T$$.

• #### Hint

Multiply the matrix $$B$$ of the form $$g$$ w.r.t. the canonical basis by the transition matrix from $$X$$ to $$K$$ from both sides to get its matrix w.r.t. the basis $$X$$.

• #### Solution

$$B_X=[id]_{XK}^T\cdot B_K \cdot [id]_{XK}$$, i.e.

$$B_X=\begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \end{pmatrix} \begin{pmatrix} 2 & 1 & 0 & 0 \\ 1 & -1 & 0 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & -1 \\ \end{pmatrix} \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \end{pmatrix} = \begin{pmatrix} 0 & 2 & 2 & 3 \\ 2 & 3 & 3 & 3 \\ 2 & 3 & 3 & 3 \\ 3 & 3 & 3 & 2 \\ \end{pmatrix}$$

The analytic expression is: $$g(u)_X = 4ab + 4ac + 6ad + 3b^2 + 6bc + 6bd + 3c^2 + 6cd +2 d^2$$ for $$[u]_X=(a, b, c, d)^T$$.

By substitution $$(3, 1, 0, 0)^T$$ for $$(a, b, c, d)^T$$ we get that $$g(u)=15$$. The same result could be obtained if we calculate the coordinates of $$u$$ w.r.t. $$K$$, i.e. $$[u]_K=(4, 4, 4, 3)^T$$.