Linear algebra question about subspaces

In summary, the conversation discusses determining whether a given set of vectors is a subspace of R3. The set consists of vectors in the form (a,b,c) where a=2b+3c. The conversation explores the definition of vector addition and scalar multiplication in R3 and how to check if the given set satisfies the requirements to be a subspace.
  • #1
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Homework Statement



This is probably a very dumb question, but I just can't wrap my head around what I'm supposed to be doing.

The question is:

"Determine whether the set is a subspace of R3:
All vectors of the form (a,b,c) where a = 2b + 3c"

Homework Equations



u + v is an element of R3
ku is an element of R3

The Attempt at a Solution



My question is.. is R3 just the set of all vectors with 3 terms..(a,b,c), (d,e,f)...etc? OR does it mean that the vector I get after multiplying by the scalar has to be of the same form as u, such that a = 2b + 3c

I think I'm phrasing this wrong so I will give an example.

Let's say I am checking vector addition.

So u = (2b + 3c, b, c) and v = (2e + 3f, e, f)

When I add them I get (2b+3c+2e+2f, b+e, c+f)

Is that in R3 since it has 3 terms which in fact would make it a vector space? Or do I need to check if the first term (2b + 3c + 2e + 2f) = 2(b+e) + 3(c+f) ?
 
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  • #2
csgirl504 said:
Is that in R3 since it has 3 terms which in fact would make it a vector space? Or do I need to check if the first term (2b + 3c + 2e + 2f) = 2(b+e) + 3(c+f) ?

Yes and yes.
It is in R³ for the reason you state. However, this only means that the given set is a subset of R³, but you are asked to show that it is a subspace, i.e. a subset which is a vector space in its own right. For that, it needs to satisfy some additional demands, for example that if you add two vectors in the subset, the result is again in the subset. Therefore, the answer to your second question is also yes.
 
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  • #3
The basic vector space is R3, the space of vectors of the form (a, b, c) where a, b, and c can be any real numbers, vector addition is defined as (a, b, c)+ (d, e, f)= (a+ d, b+ e, c+ f) and scalar multiplication is defined by x(a, b, c)= (xa, xb, xc).

The subset is the set of vectors of the form (a, b, c) where a, b, and c are numbers satisfying a= 2b+ 3c. No, it is not enough just to check that the sum of two such things is in R3, you must show they are still in that subset: if (a, b, c) and (d, e, f) are in that set, then a= 2b+ 3c and d= 2e+ 3f. The sum of the two vectors is (a+ d, b+ e, c+ f) and you must check that a+ d= (2b+ 3c)+ (2e+ 3f) is, in fact, (2b+ 2e)+ (3c+ 3f)= 2(b+ e)+ 3(c+ f).

Don't forget that you must also check scalar multpliction. x(a, b, c)= (xa, xb, xc) and now you must check that xa= x(2b+ 3c)= 2(xb)+ 3(xc).
 
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  • #4
That explains it very well! I think I was just getting a little confused. Thanks very much!
 

1. What is a subspace in linear algebra?

A subspace in linear algebra is a subset of a vector space that also satisfies all of the properties of a vector space. This means that it must contain the zero vector, be closed under vector addition and scalar multiplication, and must be non-empty.

2. How do you determine if a set of vectors is a subspace?

To determine if a set of vectors is a subspace, you must check if it satisfies all of the properties of a vector space. This includes checking if the zero vector is included, if all vectors in the set can be added together and the result is still in the set, and if all vectors can be multiplied by a scalar and the result is still in the set.

3. Can a subspace be a line or a point?

Yes, a subspace can be a line or a point. As long as the subset satisfies all of the properties of a vector space, it can be considered a subspace. In the case of a line, it would need to include the zero vector and be closed under vector addition and scalar multiplication. In the case of a point, it would only need to include the zero vector.

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

A subspace and a span are similar concepts in linear algebra, but there is a key difference. A subspace is a subset of a vector space that satisfies all of the properties of a vector space, while a span is the set of all possible linear combinations of a given set of vectors. In other words, a span is a subspace if and only if it contains all possible linear combinations of its vectors.

5. How are subspaces used in real-world applications?

Subspaces are widely used in real-world applications, particularly in fields such as physics, engineering, and computer science. They are used to model and solve problems involving linear systems, such as in analyzing linear circuits or predicting the movement of objects in space. Subspaces also have applications in data analysis and machine learning, where they are used to represent and manipulate high-dimensional data sets.

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