QuestionA square matrix is idempotent if .
Let be the vector space of all matrices with real entries. Let be the set of all idempotent matrices with real entries. Is a subspace of the vector space ?
1. Does contain the zero vector of ?
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2. Is closed under addition? If it is, enter CLOSED. If it is not, enter two matrices in whose sum is not in , using a comma separated list and syntax such as for the answer . (Hint: to show that is not closed under addition, it is sufficient to find two idempotent matrices and such that .)
3. Is closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in and a matrix in whose product is not in , using a comma separated list and syntax such as for the answer . (Hint: to show that is not closed under scalar multiplication, it is sufficient to find a real number and an idempotent matrix such that .)
4. Is a subspace of the vector space ? You should be able to justify your answer by writing a complete, coherent, and detailed proof based on your answers to parts 1-3.
choose
Studdy Solution
STEP 1
1. A subspace must contain the zero vector, be closed under addition, and be closed under scalar multiplication.
2. We are working within the vector space of all matrices with real entries.
3. The set consists of all idempotent matrices.
STEP 2
1. Check if contains the zero vector of .
2. Check if is closed under addition.
3. Check if is closed under scalar multiplication.
4. Determine if is a subspace of .
STEP 3
Check if contains the zero vector of .
The zero vector in is the zero matrix:
\[
\begin{bmatrix}
0 & 0 \\
0 & 0
\end{bmatrix}$
Check if this matrix is idempotent:
Since , the zero matrix is idempotent and belongs to .
STEP 4
Check if is closed under addition.
Consider two idempotent matrices:
Calculate :
Check if :
Since , the sum is idempotent. However, this example does not prove non-closure. Let's try another example:
Consider:
Calculate :
Check if :
This example does not show non-closure either. Let’s try:
Calculate :
Check if :
This example also shows closure. Let's try:
Calculate :
Check if :
We need a non-closure example:
Calculate :
Check if :
Thus, is not closed under addition. Example matrices are:
STEP 5
Check if is closed under scalar multiplication.
Consider a scalar and an idempotent matrix:
Calculate :
Check if :
Thus, is not closed under scalar multiplication. Example is:
STEP 6
Determine if is a subspace of .
Based on the previous steps:
- contains the zero vector of .
- is not closed under addition.
- is not closed under scalar multiplication.
Since is not closed under addition and scalar multiplication, it is not a subspace of .
The set is not a subspace of the vector space .
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