Solving W is a Subspace of R^3: Help Needed

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In summary, the problem is to determine whether W is a subspace of R^3, given that W={x:x3=2x1-x2} and x=[x1, x2, x3]. By rewriting the equation as [x1, x2, x3]= [x1, 0, 2x1- x2]= [x1, 0, 2x1]+ [0, x2, -x2], it can be seen that any vector in W can be written as a linear combination of the basis vectors [1, 0, 2] and [0, 1, -1]. Therefore, W is a subspace of R^3 and its
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justa18unlv
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I'm stuck on a problem which asks:
Determine whether W is a subspace of R^3. If W is a subspace, then give geometric description of W. The problem is W={x:x3=2x1-x2} and x=[x1, x2, x3]
I tried solving it but I'm having a hard time understanding the properties of R^n and using them. I guess I'm not as good with proof as I'm with numbers. Please help.
 
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Because x3= 2x1- x2, any vector in that subspace can be written as [x1, x2, x3]= [x1, x2, 2x1- x2]= [x1, 0, 2x1]+ [0, x2, -x2]= x1[1, 0, 2]+ x2[0, 1, -1].

Can you see what a basis for that subspace is? Of course, 2x1- x2- x3= 0 is the equation of a plane.
 

1. What is a subspace in mathematics?

A subspace in mathematics is a subset of a vector space that satisfies all of the properties of a vector space. This means that it is closed under addition and scalar multiplication, and contains the zero vector. In simpler terms, a subspace is a smaller set of vectors that still behave like a larger vector space.

2. How do you prove that a set is a subspace?

To prove that a set is a subspace, you must show that it satisfies all of the properties of a vector space. This includes closure under addition and scalar multiplication, and containing the zero vector. Additionally, you must show that the set is non-empty and has at least one vector in common with the larger vector space.

3. What is the difference between a subspace and a span?

A subspace is a subset of a vector space, while a span is a set of all linear combinations of a given set of vectors. In other words, a span is the set of all possible combinations of a set of vectors, while a subspace is a subset of those combinations that satisfies all of the properties of a vector space.

4. How do you determine if a vector is in a subspace?

To determine if a vector is in a subspace, you must check if it satisfies all of the properties of a vector space. This includes closure under addition and scalar multiplication, and containing the zero vector. If the vector satisfies all of these properties, then it is in the subspace.

5. How can knowing that W is a subspace of R^3 be helpful in solving problems?

Knowing that W is a subspace of R^3 can be helpful in solving problems because it allows you to simplify the problem by working with a smaller set of vectors. This can make calculations and proofs easier, as well as provide a better understanding of the underlying concepts. Additionally, properties of the larger vector space can often be applied to the subspace, making it easier to solve problems.

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