enter image description here

Let $\alpha,\beta,a,b,c \in \mathbb{R}.$ Consider three affine planes in the affine space $\mathbb{R}^{3}$: $P_{1}$ of equation $x+2y+\beta z=a$, $P_2$ of equation $2x+y=b$, $P_{3}$ of equation $\alpha x +(\alpha+1)y=c.$

What is the dimension of their intersection $F=P_1 \cap P_2 \cap P_3$? When F is not empty, give it as a system of equations (with the fewest possible number of equations).

my attemps

to calculate dimensional of $F$ i thing we have show that $P_1,P_2,P_3$ are affine subspace of $R^{3}$ independantes Then we can write $dim F=dim(P_1∩P_2∩P_3)=$

To show that have show that $P_1,P_2,P_3$ are affine subspace of $R^{3}$ independent we hae to show that $P_1=kerf_{1}$ with $f_1$ is a linear form non-zero and $P_2=kerf_{2}$ with $f_2$ is a linear form non-zero and $P_3=kerf_{3}$ with $f_3$ is a linear form non-zero

how we can show tha f_1 f_2 f_3 are lineare forme non zero independent

for example we can consider $P_1=Ker f_1$ with

$$\begin{array}{ccccc} f_1 & : & R^3 & \longrightarrow & R \\ & & (x,y,z) & \mapsto & f(x,y,z)=x+2y+\beta. z=a \\ \end{array}$$

we have to show $f_1$ is linear forme non zero

if $a=$ is vector space

if possible someone suggest detailed answer

any help would be appreciated!

  • $\begingroup$ The intersection is the set of points $(x,y,z)$ satisfying $$\begin{pmatrix} 1&2&\beta \\ 2&1&0 \\ \alpha&\alpha+1&0\end{pmatrix} \begin{pmatrix} x\\y\\z \end{pmatrix} = \begin{pmatrix} a\\b\\c \end{pmatrix}.$$How do you find the solution set to such an equation? $\endgroup$ – Greg Martin Dec 10 '14 at 20:40
  • 1
    $\begingroup$ i thing the easy one is Gauss–Seidel method $\endgroup$ – Educ Dec 10 '14 at 20:56
  • $\begingroup$ @GregMartin how can i explain to my friend that The intersection ? is the set of points (x,y,z) satisfying that system can you elaborate this please $\endgroup$ – Educ Dec 11 '14 at 20:41

Un point de coordonnées $(x,y,z)$ appartient à l'intersection des trois plans s'il vérifie les trois équations, c'est-à-dire le système proposé par Greg Martin. Ce système se résout facilement par la méthode de Gauss. Selon les valeurs des paramètres alpha et bêta le rang $r$ du système est différent et la dimension de $F$ est $3 -r$. Voyez http://wims.auto.u-psud.fr/wims/wims.cgi?module=U1/algebra/docsyslin.fr

L'enseignante auteur de l'exercice

A point with coordinates $(x,y,z)$ belongs to the intersection of the three planes if it satisfies the three equations, that is, the system

$$\begin{pmatrix} 1&2&\beta \\ 2&1&0 \\ \alpha&\alpha+1&0\end{pmatrix} \begin{pmatrix} x\\y\\z \end{pmatrix} = \begin{pmatrix} a\\b\\c \end{pmatrix}$$

proposed by Greg Martin. That system is easily solved by Gauß elimination. The rank $r$ of the system depends on the value of the parameters $\alpha$ and $\beta$ and the dimension of $F$ is $3-r$. See http://wims.auto.u-psud.fr/wims/wims.cgi?module=U1/algebra/docsyslin.fr

The teacher and author of the exercise.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.