As an assignment we are to calculate the dimension of the intersection of two affine subspaces $A$,$B$ in $\mathbb{R}^6$ each defined by a system of linear equations of at most 6 variables and a constant.
How does one calculate that?
I tried to stack together the equations and simplify, then parametrize the solution as a $\vec v$ + {set of parametrized vectors}.
Then the dimensionality of their intersection should be the size of the set. But when I carry out the calculations I get a contradiction.
Is my procedure valid?
How else could I arrive at the solution? What are other methods of finding the intersection?
*Edit:
Does contradiction indicate that the intersection is empty?
*Edit2: I add code I used to check if I made any numerical errors.
After running the code, result is EmptySet()
The first 3 rows of $M$ is the augumented form of $A$.
The bottom 3 rows of $M$ is the augumented form of $B$
import sympy
M = Matrix([
[3, 4, 6, 0, 0, 0, 3],
[4, 0, 0, 1, 1, 0, 2],
[0, 1, -4, 0, 0, 0, 8],
[0, 1, 1, -1, -1, 2, 0],
[0, 0, -1, 1, 1, -2, 0],
[0, 1, -7, 0, 0, 0, 0]])
x1,x2,x3,x4,x5,x6 = sympy.symbols('x1,x2,x3,x4,x5,x6')
result = sympy.linsolve(M,(x1,x2,x3,x4,x5,x6))
Best Answer
Yes: the fact that you got the output
EmptySet()indicates that the intersection of the two affine spaces is empty.Using W|A confirms this result.