### Problem StatementConsider a linear transformation $ T: \mathbb{R}^4 \rightarrow \mathbb{R}^3 $. The standard matrix for this transformation is given as:\[ \begin{bmatrix} 1 & 2 & 1 \\ -1 & 3 & 2 \\ 5 & 1 & -2 \\ -3 & -2 & 1 \end{bmatrix} \]Determine if the transformation $ T $ is a one-to-one and onto transformation. Justify your answer.

(i) If a linear transformation is one one .Then (i) If a linear transformation is onto .Then…

Answered: 1 2 -1 3 2. IfT: R → R³ is 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

### Problem Statement

Consider a linear transformation $ T: \mathbb{R}^4 \rightarrow \mathbb{R}^3 $. The standard matrix for this transformation is given as:

\[
\begin{bmatrix}
1 & 2 & 1 \\
-1 & 3 & 2 \\
5 & 1 & -2 \\
-3 & -2 & 1
\end{bmatrix}
\]

Determine if the transformation $ T $ is a one-to-one and onto transformation. Justify your answer.

Expert Solution

Step 1: Some Results:

(i) If a linear transformation $T colon R to the power of m rightwards arrow R to the power of n$ is one one .Then $K e r left parenthesis T right parenthesis equals left curly bracket 0 right curly bracket space space t h a t space i s space n u l l i t y left parenthesis T right parenthesis equals 0.$

(i) If a linear transformation $T colon R to the power of m rightwards arrow R to the power of n$ is onto .Then rank(T)=n=dim(Rⁿ)