determine whether W is a subspace of the vector space. W

rheisf 2022-01-06 Answered
determine whether W is a subspace of the vector space.
\(\displaystyle{W}={\left\lbrace{\left({x},{y}\right)}:{x}-{y}={1}\right\rbrace},{V}={R}^{{2}}\)

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

Paul Mitchell
Answered 2022-01-07 Author has 225 answers
By the definition of a subspace a nonempty subset W of a vector space is called subspace of V, if W is a vector space under the operations of addition and scalar multiplication defined in V.
From the above definition it is clear that a subspace must also be a vector space itself and thus must satisfy the axioms of a vector space.
However, the additive identity for \(\displaystyle{R}^{{2}}\) that is,(0,0) is not in the subset W .
Hence, all the vector space axiom are not satisfied by W, so it is not a vector space, and hence not a subspace.
Therefore, W is not a subspace of \(\displaystyle{R}^{{2}}\).
Not exactly what you’re looking for?
Ask My Question
0
 

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2022-01-05
determine whether W is a subspace of the vector space.
\(\displaystyle{W}={\left\lbrace{f}:{f{{\left({0}\right)}}}=-{1}\right\rbrace},{V}={C}{\left[-{1},{1}\right]}\)
asked 2022-01-04
Prove that
If W is a subspace of a vector space V and \(\displaystyle{w}_{{1}},{w}_{{2}},\ldots,{w}_{{n}}\) are in W, then \(\displaystyle{a}_{{1}}{w}_{{1}}+{a}_{{2}}{w}_{{2}}+\ldots+{a}_{{n}}{w}_{{n}}\in{W}\) for any scalars \(\displaystyle{a}_{{1}},{a}_{{2}},\ldots,{a}_{{n}}\).
asked 2022-01-05
Determine whether the set equipped with the given operations is a vector space.
For those that are not vector spaces identify the vector space axioms that fail.
The set of all real numbers x with the standard operations of addition and multiplication.
\(\displaystyle\circ\) V is not a vector space, and Axioms 7,8,9 fail to hold.
\(\displaystyle\circ\) V is not a vector space, and Axiom 6 fails to hold.
\(\displaystyle\circ\) V is a vector space.
\(\displaystyle\circ\) V is not a vector space, and Axiom 10 fails to hold.
\(\displaystyle\circ\) V is not a vector space, and Axioms 6 - 10 fail to hold.
asked 2022-01-04
Determine whether the set equipped with the given operations is a vector space.
For those that are not vector spaces identify the vector space axioms that fail.
The set of all pairs of real numbers of the form (x,0) with the standard operations on \(\displaystyle\mathbb{R}^{{2}}\).
\(\displaystyle\circ\) V is a vector space.
\(\displaystyle\circ\) V is not a vector space, and Axiom 7,8, 9 fails to hold.
\(\displaystyle\circ\) V is not a vector space, and Axioms 4 and 5 fail to hold.
\(\displaystyle\circ\) V is not a vector space, and Axioms 2 and 3 fail to hold.
\(\displaystyle\circ\) V is not a vector space, and Axiom 10 fails to hold.
asked 2022-01-06
If V is a finite dimensional vector space and W is a subspace, the W is finite dimensional. Prove it.
asked 2022-01-07
Label the following statements as being true or false.
(a) If V is a vector space and W is a subset of V that is a vector space, then W is a subspace of V.
(b) The empty set is a subspace of every vector space.
(c) If V is a vector space other than the zero vector space {0}, then V contains a subspace W such that W is not equal to V.
(d) The intersection of any two subsets of V is a subspace of V.
(e) An \(\displaystyle{n}\times{n}\) diagonal matrix can never have more than n nonzero entries.
(f) The trace of a square matrix is the product of its entries on the diagonal.
asked 2022-01-06
Let W be a subset of the vector space V where u and v are vectors in W. If (\(\displaystyle{u}\oplus{v}\)) belongs to W, then W is a subspace of V:
Select one: True or False
...