Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
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See our solution for Question 29E from Chapter 1.9 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 29E
Chapter:
Problem:
In Exercises 29 and 30, describe the possible echelon forms of the standard matrix for a linear transformation T. Use the notation of Example 1 in Section 1.2 is one-to-one. Example 1: The following matrices are in echelon form. The leading entries ( ∎ ) may have any nonzero value; the starred entries (*) may have any value (including zero). The following matrices are in reduced echelon form because the leading entries are 1’s, and there are 0’s below and above each leading 1.
Step-by-Step Solution
Given Information
We are given with a transformation: $T : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 4 }$ which is one-to-one. We have to describe the possible echelon form of the standard matrix.
Step-1:
Since mapping is from $R^3$ to $R^4$, the standard matrix is of the form $4 \times 3$. The transformation is on-to-one, if and only if the columns of A are linearly independent. Hence each columns has a pivot element. Therefore, the standard form is:
\[A = \left[ {\begin{array}{*{20}{c}} \bullet&*&*\\ 0&\bullet&*\\ 0&0&\bullet\\ 0&0&0 \end{array}} \right]\]
We are given with a transformation: $T : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 4 }$ which is one-to-one. We have to describe the possible echelon form of the standard matrix.
Step-1:
Since mapping is from $R^3$ to $R^4$, the standard matrix is of the form $4 \times 3$. The transformation is on-to-one, if and only if the columns of A are linearly independent. Hence each columns has a pivot element. Therefore, the standard form is:
\[A = \left[ {\begin{array}{*{20}{c}} \bullet&*&*\\ 0&\bullet&*\\ 0&0&\bullet\\ 0&0&0 \end{array}} \right]\]