Scalars and determining subspaces

In summary, to determine if a subset of a vector space is a subspace, it must fulfill the requirements of being closed under addition and scalar multiplication, meaning that the sum of two arbitrary vectors in the subset must also be within the subset. Additionally, there is no limitation on the scalar value as long as the resulting vector is still within the subset.
  • #1
MoreDrinks
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To determine if a subset of a vector space is a subspace, it must be closed under addition and scalar multiplication. As far as I can tell, this means adding two arbitrary vectors in the subset and having the sum be within the subset.

But...can the scalar be any number? Is there any limitation?
 
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  • #2
If ##W## is a subspace of ##V##, then ##\alpha v\in W## for any scalar ##\alpha## and for any ##w\in W##.
 
  • #3
micromass said:
If ##W## is a subspace of ##V##, then ##\alpha v\in W## for any scalar ##\alpha## and for any ##w\in W##.

Thanks, my book is terrible.
 

What are scalars and how are they different from vectors?

Scalars are single, real-valued quantities that are represented by a magnitude or numerical value only. Vectors, on the other hand, have both magnitude and direction.

What is a subspace and how is it determined?

A subspace is a subset of a vector space that satisfies all the properties of a vector space. It is determined by checking that the span of a set of vectors within the subspace is closed under addition and scalar multiplication.

Can a scalar be considered a subspace?

No, a scalar by itself cannot be considered a subspace because it does not have the necessary properties of a vector space, such as closure under addition and scalar multiplication.

How can you determine the dimension of a subspace?

The dimension of a subspace is equal to the number of linearly independent vectors within that subspace. This can be determined by reducing the vectors to row-echelon form and counting the number of non-zero rows.

What is the difference between a basis and a spanning set for a subspace?

A basis for a subspace is a set of linearly independent vectors that can be used to represent any vector within that subspace. A spanning set, on the other hand, is a set of vectors that can be used to generate all the vectors within a subspace, but they may not necessarily be linearly independent.

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