Let T: U → V be a linear transformation. Use the rank-nullity theorem to complete the information in the table below. U dim(U) rank (T) nullity (T) P4 5 Ex: 5 1 P6 Ex: 5 6 Ex: 5 Pn Ex: n+2 Ex: n+2 7
Let T: U → V be a linear transformation. Use the rank-nullity theorem to complete the information in the table below. U dim(U) rank (T) nullity (T) P4 5 Ex: 5 1 P6 Ex: 5 6 Ex: 5 Pn Ex: n+2 Ex: n+2 7
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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![Let \( T : U \to V \) be a linear transformation. Use the rank-nullity theorem to complete the information in the table below.
\[
\begin{array}{|c|c|c|c|}
\hline
U & P_4 & P_6 & P_n \\
\hline
\dim(U) & 5 & & \\
\hline
\rank(T) & & 6 & \\
\hline
\nullity(T) & 1 & & 7 \\
\hline
\end{array}
\]
**Table Explanation:**
- **\( \dim(U) \)**: Dimension of \( U \).
- \( P_4 \) has a dimension of 5.
- \( P_6 \) and \( P_n \) values are to be filled as per given expressions or calculation.
- **\( \rank(T) \)**: Rank of the transformation \( T \).
- \( P_6 \) has a rank of 6.
- **\( \nullity(T) \)**: Nullity of the transformation \( T \).
- \( P_4 \) has a nullity of 1.
- \( P_n \) has a nullity of 7.
**Expressions:**
- For \( P_6 \), \( \dim(U) \) is calculated as an example (Ex: 5).
- For \( P_6 \), \( \nullity(T) \) is also calculated as an example (Ex: 5).
- For \( P_n \), both \( \dim(U) \) and \( \rank(T) \) are calculated as expressions (Ex: n+2).
The rank-nullity theorem states:
\[
\dim(U) = \rank(T) + \nullity(T)
\]
Use this theorem to fill in the missing table values.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97fa71a9-ddb9-496b-9b0a-bf970e388fad%2F4f904492-14aa-4697-8a8d-62174c47eb49%2Fryu7ih_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( T : U \to V \) be a linear transformation. Use the rank-nullity theorem to complete the information in the table below.
\[
\begin{array}{|c|c|c|c|}
\hline
U & P_4 & P_6 & P_n \\
\hline
\dim(U) & 5 & & \\
\hline
\rank(T) & & 6 & \\
\hline
\nullity(T) & 1 & & 7 \\
\hline
\end{array}
\]
**Table Explanation:**
- **\( \dim(U) \)**: Dimension of \( U \).
- \( P_4 \) has a dimension of 5.
- \( P_6 \) and \( P_n \) values are to be filled as per given expressions or calculation.
- **\( \rank(T) \)**: Rank of the transformation \( T \).
- \( P_6 \) has a rank of 6.
- **\( \nullity(T) \)**: Nullity of the transformation \( T \).
- \( P_4 \) has a nullity of 1.
- \( P_n \) has a nullity of 7.
**Expressions:**
- For \( P_6 \), \( \dim(U) \) is calculated as an example (Ex: 5).
- For \( P_6 \), \( \nullity(T) \) is also calculated as an example (Ex: 5).
- For \( P_n \), both \( \dim(U) \) and \( \rank(T) \) are calculated as expressions (Ex: n+2).
The rank-nullity theorem states:
\[
\dim(U) = \rank(T) + \nullity(T)
\]
Use this theorem to fill in the missing table values.
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