Plz help with ques 7 6. Let T: R³ → R² be given by-LAShow that I is a linear transformation by confirming the two conditions of the definition.Ti = T7. Consider the transformation T from the problem above.(a) Find matrix A that describes the transformation T.(b) Is T an onto mapping of R³ → R²? Explain your reason.(c) Is T a 1-1 mapping? Explain your reason.(d) Describe the co-domain and range of T.(e) Do the results of b and c help you decide on part d? Describe any connections.

Since we have the linear transformation T:R3→R2 such that Tx=xy-z(a) Standard basis…

Answered: Consider the transformation T from the…

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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Plz help with ques 7

6. Let T: R³ → R² be given by
-LA
Show that I is a linear transformation by confirming the two conditions of the definition.
Ti = T
7. Consider the transformation T from the problem above.
(a) Find matrix A that describes the transformation T.
(b) Is T an onto mapping of R³ → R²? Explain your reason.
(c) Is T a 1-1 mapping? Explain your reason.
(d) Describe the co-domain and range of T.
(e) Do the results of b and c help you decide on part d? Describe any connections.

Expert Solution

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$\begin{array}{l} S i n c e w e h a v e t h e l i n e a r t r a n s f o r m a t i o n T : R^{3} \to R^{2} s u c h t h a t \\ T x = [\begin{matrix} x \\ y - z \end{matrix}] \\ (a) \\ S t a n d a r d b a s i s f o r R^{3} i s B = \{(1, 0, 0)^{t}, (0, 1, 0)^{t}, (0, 0, 1)^{t}\} a n d B a s i s f o r \\ R^{2} i s B' = \{(1, 0)^{t}, (0, 1)^{t}\} s o b y u s i n g g i v e n t r a n s f o r m a t i o n w e g e t \\ T [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}] = [\begin{matrix} 1 \\ 0 - 0 \end{matrix}] = [\begin{matrix} 1 \\ 0 \end{matrix}] = 1 [\begin{matrix} 1 \\ 0 \end{matrix}] + 0 [\begin{matrix} 0 \\ 1 \end{matrix}] \\ T [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}] = [\begin{matrix} 0 \\ 1 - 0 \end{matrix}] = [\begin{matrix} 0 \\ 1 \end{matrix}] = 0 [\begin{matrix} 1 \\ 0 \end{matrix}] + 1 [\begin{matrix} 0 \\ 1 \end{matrix}] \\ a n d T [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}] = [\begin{matrix} 0 \\ 0 - 1 \end{matrix}] = [\begin{matrix} 0 \\ - 1 \end{matrix}] = 0 [\begin{matrix} 1 \\ 0 \end{matrix}] - 1 [\begin{matrix} 0 \\ 1 \end{matrix}] \\ S i n c e w e k n o w t h a t T x = A x w h e r e A i s m a t r i x o f l i n e a r t r a n s f o r m a t i o n s o \\ T x = {[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & - 1 \end{matrix}]}_{2 \times 3} {[\begin{matrix} x \\ y \\ z \end{matrix}]}_{3 \times 1} = {[\begin{matrix} x \\ y - z \end{matrix}]}_{2 \times 1} \\ S o m a t r i x f o r T : R^{3} \to R^{2} i s A = {[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & - 1 \end{matrix}]}_{2 \times 3} \\ (b) S i n c e T : R^{n} \to R^{m} i s o n e o n e i f a n d o n l y i f r a n k o f t r a n s f o r m a t i o n m a t r i x A \\ h a s r a n k n a n d T i s o n t o i f a n d o n l y i f A h a s r a n k m . \\ W e h a v e A = {[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & - 1 \end{matrix}]}_{2 \times 3} a n d r a n k (A) = 2 \neq 3 \\ ⟹ T i s o n t o b u t n o t o n e o n e . \end{array}$