Math Problems Vault

Problem 52.1

Prove that for any $x, y \in V$ the equality holds

Q\left(a_{0}+x+y\right)=q(y)+2 f(x, y)+l(y)+Q\left(a_{0}+x\right).

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Problem 52.2

Prove that if $b=a_{0}+v(v \in V)$ is a central point of the quadratic function $Q$ , then $Q(b+x)=Q(b-x)$ for any $x \in V$ , and the linear function $y \mapsto 2 f(v, y)+l(y)$ is zero.

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Problem 52.3

Prove that the set of central points (the center) of the quadratic function $Q$ is given by the system of equations

\frac{\partial Q}{\partial x_{i}}=0 \quad (i=1, \ldots , n).

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Problem 52.4

Prove that under the transition from the affine coordinate system $\left(a_{0}, e_{1}, \ldots , e_{n}\right)$ to the coordinate system $\left(a_{0}, e_{i}^{\prime }, \ldots , e_{n}^{\prime }\right)$ given by the formula

\left[\begin{smallmatrix} x_{1} \\ \vdots \\ x_{n} \end{smallmatrix}\right]=T\left[\begin{smallmatrix} x_{1}^{\prime } \\ \vdots \\ x_{n}^{\prime } \end{smallmatrix}\right]+\left[\begin{smallmatrix} t_{1} \\ \vdots \\ t_{n} \end{smallmatrix}\right]

the matrices of the quadratic forms $Q$ and $q$ in the new coordinate system are related to their matrices in the old system by the formulas

\widetilde{A}_{Q}^{\prime }={ }^{t} \widetilde{T} \widetilde{A}_{Q} \widetilde{T}, \quad A_{q}^{\prime }={ }^{t} T A_{q} T,

where

\widetilde{T}=\left[\begin{smallmatrix} {ccc|c} & & & t_{1} \\ & T & & \vdots \\ & & & t_{n} \\ \hline 0 & \ldots & 0 & 1 \end{smallmatrix}\right]

is the matrix of the affine change of coordinates.

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Problem 52.5

Prove that the points of intersection of the affine line

x_{k}=x_{k}^{0}+r_{k} t \quad (k=1, \ldots , n)

and the quadric $Q\left(x_{1}, \ldots , x_{n}\right)=0$ are determined by the values of $t$ satisfying the equation

A t^{2}+2 B t+C=0

where

A=q(r)=\sum _{i, j=1}^{n} a_{i j} r_{i} r_{j}, \quad C=Q\left(x_{1}^{0}, \ldots , x_{n}^{0}\right)

B=\sum _{i=1}^{n} \frac{\partial Q}{\partial x_{i}}\left(x_{1}^{0}, \ldots , x_{n}^{0}\right) r_{i}=\sum _{i, j=1}^{n}\left(a_{i j} x_{j}^{0}+b_{i}\right) r_{i}.

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Problem 52.6

(a)(b)(c)(d)

Find the center of the quadratic function over the field $\mathbb {R}$ , given in some affine coordinate system:

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(a)

$2 \sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}+2 \sum_{i=1}^{n} x_{i}+1=0$ ;

(b)

$\sum_{i=1}^{n} x_{i}^{2}+2 \sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}+2 \sum_{i=1}^{n} x_{i}+1=0$ ;

(c)

$\sum_{i=1}^{n-1} x_{i} x_{i+1}+x_{1}+x_{n}+1=0$ ;

(d)

$\sum_{i=1}^{n} x_{i}^{2}+2 \sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}+x_{1}=0$ .

Problem 52.7

(a)(b)

Two quadratic functions $Q_{i}: A \rightarrow K(i=1,2)$ are called equivalent if there exists an affine transformation $f: A \rightarrow A$ such that $Q_{2}(x)=\lambda Q_{1}(f(x))$ for some $\lambda \in K^{*}$ and all $x \in A$ . Find the number of classes of equivalent quadratic functions over the field $\mathbf{Z}_{3}$ , if:

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(a)

the dimension of $A$ equals two;

(b)

the dimension of $A$ equals three.

Problem 52.8

(a)(b)

Find the number of classes of equivalent quadratic functions on an $n$ -dimensional affine space:

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(a)

over the field $\mathbb {C}$ ;

(b)

over the field $\mathbb {R}$ .

Problem 52.9

Let the point $a_{0}$ of the affine space $(A, V)$ lie on the quadric $X$ , and let the vector $u \in V$ define an asymptotic direction. Prove that the line $x=a_{0}+t u$ either lies entirely on the surface $X$ or intersects it at exactly one point.

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Problem 52.10

Let $u \in V$ be a vector of a non-asymptotic direction of the quadric $X_{Q}$ , i.e., $q(u) \neq 0$ . Prove that the midpoints of the chords of the quadric $X_{Q}$ parallel to the vector $u$ lie in a single hyperplane, and find its equation.

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Problem 52.11

(a)(b)

Prove that the direction $u$ is not asymptotic for the quadric $X$ , given by an equation in affine coordinates, and find the equation of the hyperplane conjugate to this direction:

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(a)

$u=(1,1,1,1), \quad X: x_{1} x_{2}+x_{2} x_{3}+x_{3} x_{4}-x_{1}-x_{4}=0$ ;

(b)

$u=(1,0, \ldots , 0,1), \quad \sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}+x_{1}+x_{n}=1$ .

Problem 52.12

Prove that if the center of a quadric is nonempty, then it is contained in the hyperplane conjugate to any non-asymptotic direction.

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Problem 52.13

Prove that the set of singular points of a quadric is its intersection with its own center.

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Problem 52.14

Prove that the singular points of a quadric, if they exist, form a plane, and write down its equations.

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Problem 52.15

(a)(b)(c)

Find the points of intersection of the quadric with the line:

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(a)

$x_{3}^{2}+x_{1} x_{2}-x_{2} x_{3}-5 x_{1}=0$ , $x_{1}=\frac{5-x_{2}}{3}=\frac{10-x_{3}}{7} ;$

(b)

$5 x_{1}^{2}+9 x_{2}^{2}+9 x_{3}^{2}-12 x_{1} x_{2}-6 x_{1} x_{3}+12 x_{1}-36 x_{3}=0$ , $\frac{x_{1}}{3}=\frac{x_{2}}{2}=x_{3}-4$ ;

(c)

$x_{1}^{2}-2 x_{2}^{2}+x_{3}^{2}-2 x_{1} x_{2}-x_{2} x_{3}+4 x_{1} x_{3}+3 x_{1}-5 x_{3}=0$ , $\frac{x_{1}+3}{2}=x_{2}, \quad x_{3}=0$ .

Problem 52.16

Find all lines lying on the quadric

x_{1}^{2}+x_{2}^{2}+5 x_{3}^{2}-6 x_{1} x_{2}+2 x_{2} x_{3}-2 x_{1} x_{3}-12=0

and parallel to the line

\frac{x_{1}-1}{2}=\frac{x_{2}+3}{1}=-x_{3}.

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Problem 52.17

Find the lines passing through the origin and lying on the complex quadric

x_{2}^{2}+3 x_{1} x_{2}+2 x_{2} x_{3}-x_{1} x_{3}+3 x_{1}+2 x_{3}=0

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Problem 52.18

(a)(b)

Find the equation of the quadric $Q$ after moving the origin of coordinates to the point $O^{\prime }$ :

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(a)

$Q: x_{1}^{2}+5 x_{2}^{2}+4 x_{3}^{2}+4 x_{1} x_{2}-2 x_{2} x_{3}-4 x_{1} x_{3}-2 x_{1}-10 x_{2}+4 x_{3}=0$ , $O^{\prime }=(3,0,1) ;$

(b)

$Q: x_{1}^{2}+2 x_{2}^{2}+x_{3}^{2}-4 x_{1} x_{2}+6 x_{2} x_{3}-2 x_{1} x_{3}+10 x_{1}-5=0$ , $O^{\prime }=(-1,1,2)$ .

Problem 52.19

(a)(b)(c)(d)

Find the affine type of the curve that is the intersection of the quadric and the plane:

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(a)

$3 x_{2}^{2}+4 x_{3}^{2}+24 x_{1}+12 x_{2}-72 x_{3}+360=0$ , $x_{1}-x_{2}+x_{3}=1$

(b)

$x_{1}^{2}+5 x_{2}^{2}+x_{3}^{2}+2 x_{1} x_{2}+2 x_{2} x_{3}+6 x_{1} x_{3}-2 x_{1}+6 x_{2}+2 x_{3}=0$ , $2 x_{1}-x_{2}+x_{3}=0$

(c)

$x_{1}^{2}-3 x_{2}^{2}+x_{3}^{2}-6 x_{1} x_{2}+2 x_{2} x_{3}-3 x_{2}+x_{3}-1=0$ , $2 x_{1}-3 x_{2}-x_{3}+2=0$ ;

(d)

$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-6 x_{1}-2 x_{2}+9=0$ , $x_{1}+x_{2}-2 x_{3}-1=0$ .

Problem 52.20

(a)(b)

Find the affine and metric types of the quadric given in Euclidean space $\mathbb {R}^{n+1}$ by the equations:

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(a)

$\sum_{i=1}^{n} x_{i}^{2}+\sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}+x_{1}+x_{n+1}=0 ;$

(b)

$\sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}+x_{1}+x_{2}+\ldots +x_{n}=0$ .

Problem 52.21

Determine the affine form of the quadric and find its center:

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(a)

$4 x_{1}^{2}+2 x_{2}^{2}+12 x_{3}^{2}-4 x_{1} x_{2}+8 x_{2} x_{3}+12 x_{1} x_{3}+14 x_{1}-10 x_{2}+7=0$ ;

(b)

$5 x_{1}^{2}+9 x_{2}^{2}+9 x_{3}^{2}-12 x_{1} x_{2}-6 x_{1} x_{3}+12 x_{1}-36 x_{3}=0$ ;

(c)

$5 x_{1}^{2}+2 x_{2}^{2}+2 x_{3}^{2}-2 x_{1} x_{2}-4 x_{2} x_{3}+2 x_{1} x_{3}-4 x_{2}-4 x_{3}+4=0$ ;

(d)

$x_{1}^{2}-2 x_{2}^{2}+x_{3}^{2}+6 x_{2} x_{3}-4 x_{1} x_{3}-8 x_{1}+10 x_{2}=0$ ;

(e)

$x_{1}^{2}+2 x_{1} x_{2}+x_{2}^{2}-x_{3}^{2}+2 x_{3}-1=0$ ;

(f)

$3 x_{1}^{2}+3 x_{2}^{2}+3 x_{3}^{2}-6 x_{1}+4 x_{2}-1=0$ ;

(g)

$3 x_{1}^{2}+3 x_{2}^{2}-6 x_{1}+4 x_{2}-1=0$ ;

(h)

$3 x_{1}^{2}+3 x_{2}^{2}-3 x_{3}^{2}-6 x_{1}+4 x_{2}+4 x_{3}+3=0$ ;

(i)

$4 x_{1}^{2}+x_{2}^{2}-4 x_{1} x_{2}-36=0$ ;

(j)

$x_{1}^{2}+4 x_{2}^{2}+9 x_{3}^{2}-6 x_{1}+8 x_{2}-36 x_{3}=0$ ;

(k)

$4 x_{1}^{2}-x_{2}^{2}-x_{3}^{2}+32 x_{1}-12 x_{3}+44=0$ ;

(l)

$3 x_{1}^{2}-x_{2}^{2}+3 x_{3}^{2}-18 x_{1}+10 x_{2}+12 x_{3}+14=0$ ;

(m)

$6 x_{1}^{2}+6 x_{3}^{2}+5 x_{1}+6 x_{2}+30 x_{3}-11=0$ .

Problem 52.22

Determine the metric type of the quadric in Euclidean space and determine whether it is a surface of revolution:

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(a)

$x_{3}^{2}=2 x_{1} x_{2}$ ;

(b)

$x_{3}=x_{1} x_{2}$ ;

(c)

$x_{3}^{2}=3 x_{1}+4 x_{2}$ ;

(d)

$x_{3}^{2}=3 x_{1}^{2}+4 x_{1} x_{2}$ ;

(e)

$x_{3}^{2}=x_{1}^{2}+2 x_{1} x_{2}+x_{2}^{2}+1$ ;

(f)

$x_{1}^{2}+4 x_{2}^{2}+5 x_{3}^{2}+4 x_{1} x_{2}+4 x_{3}=0$ ;

(g)

$x_{1}^{2}+2 x_{1}+3 x_{2}+4 x_{3}+5=0$ ;

(h)

$x_{3}=x_{1}^{2}+2 x_{1} x_{2}+x_{2}^{2}+1$ ;

(i)

$x_{1}^{2}-2 x_{2}^{2}+x_{3}^{2}+4 x_{1} x_{2}-8 x_{1} x_{3}-4 x_{2} x_{3}-14 x_{1}-14 x_{2}+14 x_{3}+18=0$ ;

(j)

$5 x_{1}^{2}+8 x_{2}^{2}+5 x_{3}^{2}-4 x_{1} x_{2}+8 x_{1} x_{3}+4 x_{2} x_{3}-6 x_{1}+6 x_{2}+6 x_{3}+10=0$ ;

(k)

$2 x_{1} x_{2}+2 x_{1} x_{3}+2 x_{2} x_{3}+2 x_{1}+2 x_{2}+2 x_{3}+1=0$ ;

(l)

$3 x_{1}^{2}+3 x_{2}^{2}+3 x_{3}^{2}-2 x_{1} x_{2}-2 x_{1} x_{3}-2 x_{2} x_{3}-2 x_{1}-2 x_{2}-2 x_{3}-1=0$ ;

(m)

$2 x_{1}^{2}+6 x_{2}^{2}+2 x_{3}^{2}+8 x_{1} x_{3}-4 x_{1}-8 x_{2}+3=0$ ;

(n)

$4 x_{1}^{2}+x_{2}^{2}+4 x_{3}^{2}-4 x_{1} x_{2}-8 x_{1} x_{3}+4 x_{2} x_{3}-28 x_{1}+2 x_{2}+16 x_{3}+45=0$ ;

(o)

$2 x_{1}^{2}+5 x_{2}^{2}+2 x_{3}^{2}-2 x_{1} x_{2}-4 x_{1} x_{3}+2 x_{2} x_{3}+2 x_{1}-10 x_{2}-2 x_{3}-1=0$ ;

(p)

$7 x_{1}^{2}+7 x_{2}^{2}+16 x_{3}^{2}-10 x_{1} x_{2}-8 x_{1} x_{3}-8 x_{2} x_{3}-16 x_{1}-16 x_{2}-8 x_{3}+72=0$ ;

(q)

$4 x_{1}^{2}+4 x_{2}^{2}-8 x_{3}^{2}-10 x_{1} x_{2}+4 x_{1} x_{3}+4 x_{2} x_{3}-16 x_{1}-16 x_{2}+10 x_{3}-2=0$ ;

(r)

$2 x_{1}^{2}-7 x_{2}^{2}-4 x_{3}^{2}+4 x_{1} x_{2}-16 x_{1} x_{3}+20 x_{2} x_{3}+$

+60 x_{1}-12 x_{2}+12 x_{3}-90=0 ;

(s)

$2 x_{1} x_{2}+2 x_{1} x_{3}-2 x_{1} x_{4}-2 x_{2} x_{3}+2 x_{2} x_{4}+$

+2 x_{3} x_{4}-2 x_{2}-4 x_{3}-6 x_{4}+5=0 ;

(t)

$3 x_{1}^{2}+3 x_{2}^{2}+3 x_{3}^{2}+3 x_{4}^{2}-2 x_{1} x_{2}-2 x_{1} x_{3}-2 x_{1} x_{4}-$

-2 x_{2} x_{3}-2 x_{2} x_{4}-2 x_{3} x_{4}=36

Problem 52.23

For which values of the parameter $a$ is the quadric

x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+2 a x_{1} x_{2}+2 a x_{1} x_{3}+2 a x_{2} x_{3}=4 a

an ellipsoid?

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Problem 52.24

Under what necessary and sufficient condition do two hyperboloids have a common asymptotic cone?

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Problem 52.25

Find the affine and metric types of the quadric given in Euclidean space $\mathbb {R}^{n+1}$ by the equation

a \sum _{i=1}^{n} x_{i}^{2}+2 b \sum _{1 \leqslant i<j \leqslant n} x_{i} x_{j}+2 c \sum _{i=1}^{n+1} x_{i}=0

depending on the values of the parameters $a, b$ , and $c$ .

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Problem 52.26

(a)(b)(c)

A quadric is called $k$ -planar if through any of its points there passes at least one $k$ -dimensional plane lying entirely on the quadric, but no $(k+1)$ -dimensional plane is contained in the quadric. Prove that:

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(a)

a quadric of type $\mathrm{I}_{n, s}^{\prime }$ over $\mathbb {R}$ $k$ -planar, where $k=\min (s, n-s)$ ;

(b)

a nonsingular quadric of type $\mathrm{I}_{n, s}$ over $\mathbb {R}(s-1)$ -planar, if $0 \leqslant s \leqslant n / 2$ , and $(n-s)$ -planar, if $s>n / 2$ ;

(c)

a nonsingular quadric of type $\mathrm{II}_{n, s}$ over $\mathbb {R} s$ -planar, if $0 \leqslant s \leqslant n / 2$ , and $(n-1-s)$ -planar, if $s>n / 2$ .

Problem 52.27

Determine for which values of the parameters $a, b, c \neq 0$ , on the quadric

a \sum _{i=1}^{n} x_{i}^{2}+2 b \sum _{1 \leqslant i<j \leqslant n} x_{i} x_{j}+2 c \sum _{i=1}^{n+1} x_{i}=0

in the space $\mathbb {R}^{n+1}$ , there lies a plane of largest dimension, and find the dimension of this plane.

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Problem 52.28

Let $\left(e_{1}, \ldots , e_{n}\right)$ be a basis of a vector space $V$ over a field $K$ of characteristic different from 2.

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(a)

Prove that for $n=4$ all decomposable elements $v_{1} \wedge v_{2}$ in $\Lambda^{2} V$ satisfy a nonsingular homogeneous quadratic equation $Q\left(x_{0}, \ldots , x_{5}\right)=0$ (the Plücker quadric).

(b)

Prove that all decomposable vectors in the space $\bigwedge^{r} V$ , $2 \leqslant r \leqslant n-2$ , satisfy a system of homogeneous quadratic equations

Q_{i}\left(x_{0}, \ldots , x_{\binom {n}{r}-1}\right)=0

(c)

Suppose the space $V$ carries a nonsingular quadratic form $Q$ . Then on the space $\bigwedge^{p} V$ one can introduce a quadratic form $Q^{(p)}$ by the formulas

\begin{gathered} Q^{(0)}=1, \\ Q^{(p)}\left(v_{1} \wedge \ldots \wedge v_{p}\right)=\operatorname {det}\left[\begin{smallmatrix} Q\left(v_{1}, v_{1}\right) & \ldots & Q\left(v_{1}, v_{p}\right) \\ \ldots \ldots \ldots \ldots \ldots & \ldots & \ldots \\ Q\left(v_{p}, v_{1}\right) & \ldots & Q\left(v_{p}, v_{p}\right) \end{smallmatrix}\right]. \end{gathered}

Prove that the resulting extension of the form $Q$ to the algebra $\wedge (V)$ is a nonsingular quadratic form on $\wedge (V)$ .

(d)

An orientation of an $n$ -dimensional vector space with a nonsingular quadratic form $Q$ is an element $d \in \Lambda^{n} V$ for which $Q^{(n)}(d)=1$ . Prove that if $\operatorname {det} Q$ is a square in the field $K$ , then $V$ has exactly two orientations, and for either of them (say, $d$ ) one can define an isomorphism of vector spaces $\lambda_{d}: V \rightarrow \bigwedge^{n-1} V$ satisfying the relation

v \wedge x=Q\left(v, \lambda _{d}^{-1} x\right) d=Q^{(n-1)}\left(\lambda _{d} v, x\right) d, \quad v \in \bigwedge ^{n-1} V, x \in V

( $Q$ is the bilinear form corresponding to the quadratic form $Q$ ).

(e)

Using the isomorphism $\lambda_{d}$ from the previous part, define, in the case $\operatorname {dim} V=3$ , a bilinear map $V \times V \rightarrow V$ by the formula

[x, y]=\lambda _{d}^{-1}(x \wedge y), \quad x, y \in V.

Prove that the multiplication in $V$ defined in this way endows $V$ with the structure of a Lie algebra over $K$ .