Let V, W, and Z be vector spaces, and let

b2sonicxh 2022-01-07 Answered
Let V, W, and Z be vector spaces, and let \(\displaystyle{T}:{V}\rightarrow{W}\) and \(\displaystyle{U}:{W}\rightarrow{Z}\) be linear.
If U and T are one-to-one and onto, prove that UT is also

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

Navreaiw
Answered 2022-01-08 Author has 3552 answers
Let U and T is one to one.
Assume, \(\displaystyle{U}{T}{\left({x}\right)}={U}{T}{\left({y}\right)}\)
\(\displaystyle{T}{\left({x}\right)}={T}{\left({y}\right)}\) U is one to one
\(\displaystyle{x}={y}\) T is one to one
So, if U and T is one to one, then UT is also one to one.
Suppose U and T is onto, then by definition of onto \(\displaystyle{T}{\left({x}\right)}={y}\), for all y W and
\(\displaystyle{U}{\left({y}\right)}={z}\) for all \(\displaystyle{z}\in{Z}\).
\(\displaystyle{U}{T}{\left({x}\right)}={U}{T}{\left({y}\right)}\)
\(\displaystyle={2}\)
So, that UT is onto.
Hence, UT is onto.
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
Let V, W, and Z be vector spaces, and let \(\displaystyle{T}:{V}\rightarrow{W}\) and \(\displaystyle{U}:{W}\rightarrow{Z}\) be linear.
If UT is onto, prove that U is onto.Must T also be onto?
asked 2022-01-04
Let V, W, and Z be vector spaces, and let \(\displaystyle{T}:{V}\rightarrow{W}\) and \(\displaystyle{U}:{W}\rightarrow{Z}\) be linear.
If UT is one-to-one, prove that T is one-to-one. must U also be one-to-one?
asked 2022-01-04
Let V and W be vector spaces, let \(\displaystyle{T}:{V}\rightarrow{W}\) be linear, and let \(\displaystyle{\left\lbrace{w}_{{1}},{w}_{{2}},\ldots,{w}_{{k}}\right\rbrace}\) be a linearly independent set of k vectors from R(T). Prove that if \(\displaystyle{S}={\left\lbrace{v}_{{1}},{v}_{{2}},...,{v}_{{k}}\right\rbrace}\) is chosen so that \(\displaystyle{T}{\left({v}_{{i}}\right)}={W}_{{i}}\) for \(\displaystyle{i}={1},{2},\ldots,{k},\) then S is linearly independent.
asked 2022-01-05
Let V and W be vector spaces and \(\displaystyle{T}:{V}\rightarrow{W}\) be linear. Let \(\displaystyle{\left\lbrace{y}_{{1}},…,{y}_{{k}}\right\rbrace}\) be a linearly independent subset of \(\displaystyle{R}{\left({T}\right)}\). If \(\displaystyle{S}={\left\lbrace{x}_{{1}},…,{x}_{{k}}\right\rbrace}\) is chosen so that \(\displaystyle{T}{\left(_\xi\right)}={y}_{{i}}\) for \(\displaystyle{i}={1},…,{k}\), prove that S is linearly independent.
asked 2022-01-07
Let V and W be vector spaces, and let T and U be nonzero linear transformations from V into W. If R(T) ∩ R(U) = {0}, prove that {T, U} is a linearly independent subset of L(V, W).
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
asked 2021-02-25

Let U and W be vector spaces over a field K. Let V be the set of ordered pairs (u,w) where \(u \in U\) and \(w \in W\). Show that V is a vector space over K with addition in V and scalar multiplication on V defined by
\((u,w)+(u',w')=(u+u',w+w')\ and\ k(u,w)=(ku,kw)\)
(This space V is called the external direct product of U and W.)

...