Question

# Assume that T is a linear transformation. Find the standard matrix of T.T=RR^2rarrRR^4 such that T(e_1)=(7,1,7,1),and T(e_2)=(-8,5,0,0),where e_1=(1,0), and e_2=(0,1).

Alternate coordinate systems

Assume that T is a linear transformation. Find the standard matrix of T.
$$\displaystyle{T}=\mathbb{R}^{{2}}\rightarrow\mathbb{R}^{{4}}\ {s}{u}{c}{h} \ {t}\hat \ {{T}}{\left({e}_{{1}}\right)}={\left({7},{1},{7},{1}\right)},{\quad\text{and}\quad}{T}{\left({e}_{{2}}\right)}={\left(-{8},{5},{0},{0}\right)},{w}{h}{e}{r}{e}{\ e}_{{1}}={\left({1},{0}\right)},{\quad\text{and}\quad}{e}_{{2}}={\left({0},{1}\right)}$$.

2020-12-29

Let $$\displaystyle{T}:{V}\rightarrow{W}$$ be a linear transformation. Suppose dim $$\displaystyle{V}={n},{\quad\text{and}\quad}{S}={\left\lbrace{v}_{{1}},\ldots,{v}_{{n}}\right\rbrace}$$ is an ordered basis for V and suppose dim $$\displaystyle{W}={m}{\quad\text{and}\quad}{B}={\left\lbrace{w}_{{1}},\ldots,{w}_{{m}}\right\rbrace}$$ is an ordered basis for W
1.Calculate $$\displaystyle{T}{\left({v}_{{1}}\right)},{T}{\left({v}_{{2}}\right)},\ldots,{T}{\left({v}_{{n}}\right)}$$
2.Find the coordinate vectors $$\displaystyle{\left({T}{\left({v}_{{1}}\right)}\right)}_{{B}}{\mid}{\left({T}{\left({v}_{{2}}\right)}\right)}_{{B}},\ldots,{\left({T}{\left({v}_{{m}}\right)}\right)}_{{B}}$$
3.Write the matrix with columns as the column vectors calculated in Step 2:
$$\displaystyle{M}={\left[{\left({T}{\left({v}_{{1}}\right)}\right)}_{{B}}{\left|{\left({T}{\left({v}_{{2}}\right)}\right)}_{{B}}\right|}\ldots{\mid}{\left({T}{\left({v}_{{m}}\right)}\right)}_{{B}}\right]}$$
Clearly, here $$\displaystyle{V}=\mathbb{R}^{{2}}{\quad\text{and}\quad}{W}=\mathbb{R}^{{4}}.{H}{e}{r}{e}{S}={\left\lbrace{e}_{{1}}={\left({1},{0}\right)},{e}_{{2}}={\left({0},{1}\right)}\right\rbrace}$$is the standard ordered basis for $$\displaystyle{V}{\quad\text{and}\quad}{B}={\left\lbrace\epsilon_{{1}}={\left({1},{0},{0},{0}\right)},\epsilon_{{2}}={\left({0},{1},{0},{0}\right)},\epsilon_{{3}}={\left({0},{0},{0},{1}\right)},\epsilon_{{4}}={\left({0},{0},{0},{1}\right)}\right\rbrace}$$ is the standard ordered basis for W.
Now we calculate $$[T(e_1)]_B,[T(e_2)]_B. \ Since \ T(e_1)=(7,1,7,1)\ and \ T(e_2)= (-8,5,0,0)$$ it follows that
$$\displaystyle{T}{\left({e}_{{1}}\right)}={7}\epsilon_{{1}}+{1}\epsilon_{{2}}+{7}\epsilon_{{3}}+{1}\epsilon_{{4}}\Rightarrow{\left[{T}{\left({e}_{{1}}\right)}\right]}_{{B}}={\left({7},{1},{7},{1}\right)}$$
$$\displaystyle{T}{\left({e}_{{2}}\right)}={8}\epsilon_{{1}}+{5}\epsilon_{{2}}+{0}\epsilon_{{3}}+{0}\epsilon_{{4}}\Rightarrow{\left[{T}{\left({e}_{{2}}\right)}\right]}_{{B}}={\left(-{8},{5},{0},{0}\right)}$$
Now we create the matrix M with columns as the column as the column vectors of $$\displaystyle{\left[{T}{\left({e}_{{1}}\right)}\right]}_{{B}},{\left[{T}{\left({e}_{{2}}\right)}\right]}_{{B}}$$.
Therefore
$$\displaystyle{M}={\left[{T}\right]}={\left[\begin{array}{cc} {7}&-{8}\\{1}&{5}\\{7}&{0}\\{1}&{0}\end{array}\right]}$$