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de_brook
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Is there a linear space V in which the union of any subspaces of V is a subspace except the trivial subspaces V and {0}? pls help
yyat said:A vector space V can only have non-trivial subspaces if [tex]\dim V\ge 2[/tex].
This means you can choose two linearly independent vectors u, w, which generate 1-dimensional subspaces U, W respectively. Can [tex]U\cup W[/tex] be a subspace? Hint: try to find a linear combination of u,w that is not in [tex]U\cup W[/tex].
de_brook said:I have tried searching for such spaces but i could only find for spaces whose dimension is less than 2.
yyat said:[tex]\dim\mathbb{R}^n=n[/tex], surely you knew that?
yyat said:[tex]\dim\mathbb{R}^n=n[/tex], surely you knew that?
yyat said:[tex]\dim\mathbb{R}^n=n[/tex], surely you knew that?
ThirstyDog said:I think the short answer is No.
ThirstyDog said:If the dimension of the space is less than two then the only subspace are V and {0} as yyat pointed out. Hence your question is answered in this case.
If the dimension of the space is greater or equal to two then consider spaces X and Y generated by linearly independent vectors x and y. x+y does not belong to [tex] X \Cup Y [/tex]. Implying you can't pick any subspaces and the union will be a subspace.
A union of subspaces in a linear space is the set of all vectors that can be written as a linear combination of vectors from each individual subspace. This means that any vector in the union can be expressed as a sum of vectors from each subspace.
A direct sum of subspaces is a special case of a union, where the intersection of the subspaces is only the zero vector. In a union of subspaces, the intersection can be any subspace, including the trivial subspace of just the zero vector.
The dimension of a union of subspaces is the sum of the dimensions of each individual subspace, minus the dimension of their intersection. This can be written as: dim(U + V) = dim(U) + dim(V) - dim(U ∩ V).
A union of subspaces can be represented geometrically as the space spanned by all the vectors in each individual subspace. This can be visualized as the combination of all the lines, planes, or higher-dimensional subspaces that make up each individual subspace.
Unions of subspaces have various applications in fields such as engineering, physics, and computer science. For example, in robotics, a union of subspaces can represent the set of all possible positions and orientations of a robot's end effector. In computer graphics, a union of subspaces can represent the space of all possible images that can be formed by combining different textures. Additionally, in machine learning, unions of subspaces can be used to model complex data sets with multiple underlying factors.