Assume that T is a linear transformation. Find the standard matrix of T. T:R2→R4,T(e1)=(3,1,3,1) and T(e2)=(−5,2,0,0), where e1=(1,0) and e2=90,1)

We know, that e1=[10] e2=[01] The task is given T(e1)=[3131] T(e2)=[−5200] There exists a unique matrix A for the linear transformation T for which it holds T(u)=Au for all u and A is a form A=[T(e1),…,T(en)], where ei,i=1,2,… are vectors from the identity matrix, respectively to columns. Matrix A is the standart matrix for the linear transformation T. So, the standart form is: A=[T(e1)T(e2)]=[3−5123010] A=[3−5123010]

"Assume that T is a linear transformation. Find..." was answered by our community

Priscilla Johnston

Answered question

2022-01-06

Assume that T is a linear transformation. Find the standard matrix of T.

T : R^{2} \to R^{4}, T (e_{1}) = (3, 1, 3, 1)

and

T (e_{2}) = (- 5, 2, 0, 0)

, where

e_{1} = (1, 0)

and

e_{2} = 90, 1)

Answer & Explanation

SlabydouluS62

Skilled2022-01-07Added 52 answers

We know, that

e_{1} = [\begin{matrix} 1 \\ 0 \end{matrix}]

e_{2} = [\begin{matrix} 0 \\ 1 \end{matrix}]

The task is given

T (e_{1}) = [\begin{matrix} 3 \\ 1 \\ 3 \\ 1 \end{matrix}]

T (e_{2}) = [\begin{matrix} - 5 \\ 2 \\ 0 \\ 0 \end{matrix}]

There exists a unique matrix A for the linear transformation T for which it holds

T (u) = A u

for all u and A is a form

A = [T (e_{1}), \dots, T (e_{n})]

, where

e_{i}, i = 1, 2, \dots

are vectors from the identity matrix, respectively to columns.
Matrix A is the standart matrix for the linear transformation T.
So, the standart form is:

A = [T (e_{1}) T (e_{2})] = [\begin{matrix} 3 & - 5 \\ 1 & 2 \\ 3 & 0 \\ 1 & 0 \end{matrix}]

A = [\begin{matrix} 3 & - 5 \\ 1 & 2 \\ 3 & 0 \\ 1 & 0 \end{matrix}]

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