How can be proved this?

Augustus Acevedo
2022-07-01
Answered

if ${e}_{1},\dots ,{e}_{n}$ is a complete set of orthogonal idempotents in a commutative ring, then $R=R{e}_{1}\times \cdots \times R{e}_{n}$ is a direct product decomposition.

How can be proved this?

How can be proved this?

You can still ask an expert for help

Shawn Castaneda

Answered 2022-07-02
Author has **17** answers

Define a map $f:R\to R{e}_{1}\times \cdots \times R{e}_{n}$ by $f(x)=(x{e}_{1},\dots ,x{e}_{n})$ and prove that $f$ is a surjective ring homomorphism with $\mathrm{ker}f=(0)$. (Note that $R{e}_{i}$ are unitary commutative rings and $f$ is a homomorphism of unitary rings.)

Willow Pratt

Answered 2022-07-03
Author has **5** answers

You have ${1}_{R}=\sum _{i=1}^{n}{e}_{i}$ and ${e}_{i}{e}_{j}={\delta}_{ij}{e}_{i}$ for each $i,j.$. From here on in, it is just a matter of writing down consequences of that.

asked 2021-05-27

g is related to one of the six parent functions. (a) Identify the parent function f. (b) Describe the sequence of transformations from f to g. (c) Sketch the graph of g by hand. (d) Use function notation to write g in terms of the parent function f.

asked 2022-06-17

Consider a linear transformation $T:{\mathbb{R}}^{3}\to {\mathbb{R}}^{3}$ satisfying

$T\left(\begin{array}{c}1\\ 2\\ 3\end{array}\right)=\left(\begin{array}{c}4\\ 5\\ 6\end{array}\right)\text{}\text{and}\text{}T\left(\begin{array}{c}2\\ 3\\ 4\end{array}\right)=\left(\begin{array}{c}5\\ 6\\ 7\end{array}\right).$The question told me to find $T\left(\begin{array}{c}3\\ 4\\ 5\end{array}\right)$ which I did but I am wondering how would I find the transformation matrix $T$?

$T\left(\begin{array}{c}1\\ 2\\ 3\end{array}\right)=\left(\begin{array}{c}4\\ 5\\ 6\end{array}\right)\text{}\text{and}\text{}T\left(\begin{array}{c}2\\ 3\\ 4\end{array}\right)=\left(\begin{array}{c}5\\ 6\\ 7\end{array}\right).$The question told me to find $T\left(\begin{array}{c}3\\ 4\\ 5\end{array}\right)$ which I did but I am wondering how would I find the transformation matrix $T$?

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$n\times n$ 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.

(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

(f) The trace of a square matrix is the product of its entries on the diagonal.

asked 2021-05-26

Refer to the system of linear equations

asked 2021-05-02

h is related to one of the six parent functions. (a) Identify the parent function f. (b) Describe the sequence of transformations from f to h. (c) Sketch the graph of h by hand. (d) Use function notation to write h in terms of the parent function f.

asked 2022-06-23

Consider the matrix transformation $T:{\mathbb{R}}^{3}\to {\mathbb{R}}^{3}$ defined by

$T(x,y,z)=(\u20134x+3y+z,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2x\u20135y,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}6y+7z).$

Find the standard matrix for $T$, and use that matrix to find $T(1,\u20133).$.

$T(x,y,z)=(\u20134x+3y+z,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2x\u20135y,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}6y+7z).$

Find the standard matrix for $T$, and use that matrix to find $T(1,\u20133).$.

asked 2022-05-22

If $T\in \text{End}(V)$ such that $T({x}_{1})=2{x}_{1}+{x}_{2}$ and $T({x}_{2})={x}_{1}$, and ${y}_{1}=4{x}_{1}+2{x}_{2}$ and ${y}_{2}={x}_{1}-{x}_{2}$ , determine the matrix $T$ with respect to the basis $\{{x}_{1},{x}_{2}\}$ and with respect to the new basis $\{{y}_{1},{y}_{2}\}$. Furthermore, it is possible to find an invertible matrix $P$ such that ${P}^{-1}AP=B$, where $A$ is the matrix transformation with respect to the basis $\{{x}_{1},{x}_{2}\}$ and $B$ is the matrix transformation with respect to the basis $\{{y}_{1},{y}_{2}\}$.

For the first part, I know I need to find some matrix $D={C}^{-1}AC$ , such that $A$ is the transformation matrix with respect to the standard basis, and $C$ is the change of basis matrix, but I am unsure how to construct $C$ and thus ${C}^{-1}$. The transformation matrix for $T$ is:

$A=\left[\begin{array}{cc}2& 1\\ 1& 0\end{array}\right]$

For the first part, I know I need to find some matrix $D={C}^{-1}AC$ , such that $A$ is the transformation matrix with respect to the standard basis, and $C$ is the change of basis matrix, but I am unsure how to construct $C$ and thus ${C}^{-1}$. The transformation matrix for $T$ is:

$A=\left[\begin{array}{cc}2& 1\\ 1& 0\end{array}\right]$