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## Main Question or Discussion Point

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

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- #1

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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

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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].

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I have tried searching for such spaces but i could only find for spaces whose dimension is less than 2.

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].

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[tex]\dim\mathbb{R}^n=n[/tex], surely you knew that?I have tried searching for such spaces but i could only find for spaces whose dimension is less than 2.

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yyat; Do you mean i can obtain a linear subspace V of \mathbb{R}^n such that the union of any subspaces of V is a subspace of V?[tex]\dim\mathbb{R}^n=n[/tex], surely you knew that?

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yyat; Do you mean i can obtain a linear subspace V of [tex]\mathbb{R}^n such that the union of any subspaces of V is a subspace of V?[tex]\dim\mathbb{R}^n=n[/tex], surely you knew that?

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yyat; Do you mean i can obtain a linear subspace V of [tex]\mathbb{R}^n[/tex], such that the union of any subspaces of V is a subspace of V?[tex]\dim\mathbb{R}^n=n[/tex], surely you knew that?

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I think the short answer is No.

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what about if you are not working n dimensional space, can you still find such a space

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what about if you are not working with n-dimensional space, can you still find such a space?I think the short answer is No.

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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.

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thanks so much.

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.

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