One more ? proving subspaces

Thread starter chantella28
Start date Nov 4, 2005
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Subspaces

In summary, a subspace is a subset of a vector space that satisfies the three conditions of closure under addition, closure under scalar multiplication, and contains the zero vector. To prove a subset is a subspace, these three conditions must be satisfied, which can be done through direct proof or using the subspace test. The main difference between a subspace and a vector space is that a subspace is a subset of a vector space, while a vector space is a set of vectors that satisfies certain properties. A subspace can contain only one vector and can also be a line or a plane, as long as the three conditions are met.

Nov 4, 2005

chantella28

see attached equation

thank you!

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Nov 4, 2005

HallsofIvy

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What is the definition of "subspace". Does this set satisfy each of the conditions in that definition?

1. What is a subspace?

A subspace is a subset of a vector space that satisfies the three conditions of closure under addition, closure under scalar multiplication, and contains the zero vector. In other words, it is a set of vectors that can be added together and multiplied by scalars to produce vectors that are also in the subspace.

2. How do you prove a subset is a subspace?

To prove that a subset is a subspace, you must show that it satisfies the three conditions of closure under addition, closure under scalar multiplication, and contains the zero vector. This can be done through direct proof, where you show that the subset satisfies each condition, or by using the subspace test, which checks if the subset contains the zero vector and is closed under addition and scalar multiplication.

3. What is the difference between a subspace and a vector space?

A subspace is a subset of a vector space, while a vector space is a set of vectors that satisfies a list of properties, including closure under addition and scalar multiplication, associativity, and distributivity. In other words, all vector spaces are subspaces, but not all subspaces are vector spaces.

4. Can a subspace contain only one vector?

Yes, a subspace can contain only one vector. As long as this vector satisfies the three conditions of closure under addition, closure under scalar multiplication, and contains the zero vector, it can be considered a subspace.

5. Can a subspace be a line or a plane?

Yes, a subspace can be a line or a plane. In fact, a line or a plane is often used to represent a subspace in 2D or 3D space. As long as the line or plane satisfies the three conditions of closure under addition, closure under scalar multiplication, and contains the zero vector, it can be considered a subspace.