Let T : R4 → R³ be a linear transformation such thatT(1,0,0,0) = (3, 2, 1), T(0,1,0,0) =(-2,1,7),T(0,0, 1, 0) = (0, 1,0), T(0,0,0,1) = (2, –1,0)(a) Find the standard matrix for this linear transformation.(b) Find T(3, 2, 1, 4).

Answered: Let T : R* → R³ be a linear…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let T : R4 → R³ be a linear transformation such that
T(1,0,0,0) = (3, 2, 1), T(0,1,0,0) =(-2,1,7),
T(0,0, 1, 0) = (0, 1,0), T(0,0,0,1) = (2, –1,0)
(a) Find the standard matrix for this linear transformation.
(b) Find T(3, 2, 1, 4).

Expert Solution

Step 1

We have:

$T (1, 0, 0, 0) = (3, 2, 1) T (0, 1, 0, 0) = (- 2, 1, 7) T (0, 0, 1, 0) = (0, 1, 0) T (0, 0, 0, 1) = (2, - 1, 0)$

Now, if we have:

$e_{1} = (\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}), e_{2} = (\begin{matrix} 0 \\ 1 \\ 0 \\ 0 \end{matrix}), e_{3} = (\begin{matrix} 0 \\ 0 \\ 1 \\ 0 \end{matrix}), e_{4} = (\begin{matrix} 0 \\ 0 \\ 0 \\ 1 \end{matrix})$

Then we are given:

$T (e_{1}) = T (\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}) = (\begin{matrix} 3 \\ 2 \\ 1 \end{matrix}) T (e_{2}) = T (\begin{matrix} 0 \\ 1 \\ 0 \\ 0 \end{matrix}) = (\begin{matrix} - 2 \\ 1 \\ 7 \end{matrix}) T (e_{3}) = T (\begin{matrix} 0 \\ 0 \\ 1 \\ 0 \end{matrix}) = (\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}) T (e_{4}) = T (\begin{matrix} 0 \\ 0 \\ 0 \\ 1 \end{matrix}) = (\begin{matrix} 2 \\ - 1 \\ 0 \end{matrix})$