Proving W is a Subspace of V: Let u & v be Vectors in V

In summary, the conversation discusses how to prove that a set of all linear combinations of two fixed vectors in a vector space is a subspace of that vector space. The method involves taking two vectors from the set and showing that their linear combination is also in the set.
  • #1
ECE
7
0
Let u and v be (fixed) vectors in the vector space V. Show that the set W of all linear combinations au+bv of u and v is a subspace of V.

I cannot prove the above proof properly. Can anyone help.

-Thanks
 
Physics news on Phys.org
  • #2
ECE said:
Let u and v be (fixed) vectors in the vector space V. Show that the set W of all linear combinations au+bv of u and v is a subspace of V.

I cannot prove the above proof properly. Can anyone help.

-Thanks

Take two vectors from W, and show that their linear combination is also in W.
 
  • #3
"Take two vectors from W" means taking two linear combinations, perhaps with different "a" and "b', say au+ bv and cu+ dv. "Their linear combination" would be something like x(au+ bv)+ y(cu+ dv) for numbers, x and y. "Show it is also in W" means "show it satisfies the definition of W". Here that means show that it also can be written au+ bv for some choices of a and b.
 
  • #4
Thanks i understand it now
 

1. What is a subspace?

A subspace is a subset of a vector space that satisfies the three properties of a vector space: closure under addition, closure under scalar multiplication, and contains the zero vector.

2. How do you prove that W is a subspace of V?

To prove that W is a subspace of V, you need to show that it satisfies the three properties of a vector space: closure under addition, closure under scalar multiplication, and contains the zero vector. You also need to prove that W is a subset of V.

3. What is the significance of u and v being vectors in V?

U and v being vectors in V means that they are elements of the vector space V. This is important because to prove that W is a subspace of V, you need to show that it satisfies the properties of a vector space for all possible vectors in V, including u and v.

4. Can you give an example of a subspace of a vector space?

One example of a subspace is the set of all 2x2 matrices with real number entries. This set satisfies the three properties of a vector space and is a subset of the vector space of all 2x2 matrices.

5. Why is it important to prove that W is a subspace of V?

Proving that W is a subspace of V is important because it allows us to understand the structure and properties of the vector space V. It also helps us to identify and work with smaller, more manageable subsets of V.

Similar threads

I Vector subspace and basis vectors in the context of data science
    • Linear and Abstract Algebra
Replies
6
Views
869
I If T is diagonalizable then is restriction operator diagonalizable?
    • Linear and Abstract Algebra
Replies
3
Views
1K
I Question from a proof in Axler 2nd Ed, 'Linear Algebra Done Right'
    • Linear and Abstract Algebra
Replies
3
Views
1K
I Showing that a set of differentiable functions is a subspace of R
    • Linear and Abstract Algebra
Replies
19
Views
4K
I Terminologies used to describe tensor product of vector spaces
    • Linear and Abstract Algebra
Replies
7
Views
238
MHB Equivalence of Subspace Statements
    • Linear and Abstract Algebra
Replies
9
Views
983
I Smallest subspace if a plane and a line are passing through the origin
    • Linear and Abstract Algebra
Replies
2
Views
1K
I Trying to get a better understanding of the quotient V/U in linear algebra
    • Linear and Abstract Algebra
Replies
10
Views
2K
MHB Show that φ(v)=λv for a vector v and a coefficient λ
    • Linear and Abstract Algebra
Replies
1
Views
836
I A = norm-preserving linear map (+other conditions) => A = lin isometry
    • Linear and Abstract Algebra
Replies
4
Views
865

Hot Threads

Recent Insights

Back
Top