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Why does the 2x2 matrix with a trace equal to 1 not contain any zero vectors?
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The trace of a square n×n matrix A=(aij) is the sum a11+a22+⋯+ann of the entries on its main diagonal.
Let V be the vector space of all 2×2 matrices with real entries. Let H be the set of all 2×2 matrices with real entries that have trace 1. Is H a subspace of the vector space V?
Does H contain the zero vector of V?
H does not contain the zero vector of V
Hello, I was reviewing my homework problems and I can't seem to understand the logic behind these correct answers.
The matrix <[1, 0], [0, 0]> has a trace of 1, is 2x2, and uses real entries while having a zero vector. Why is the answer 'does not contain a zero vector'?
Does the zero vector mean zero's in the entire matrix?
I was under the impression that a matrix can be broken up into n vectors so [1,0] is one vector and [0,0] is another vector thereby meaning that there is a zero vector.
linear-algebra vector-spaces vectors matrix-calculus
asked 1 hour ago
S. Snake
163
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S. Snake is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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up vote
3
down vote
favorite
The trace of a square n×n matrix A=(aij) is the sum a11+a22+⋯+ann of the entries on its main diagonal.
Let V be the vector space of all 2×2 matrices with real entries. Let H be the set of all 2×2 matrices with real entries that have trace 1. Is H a subspace of the vector space V?
Does H contain the zero vector of V?
H does not contain the zero vector of V
Hello, I was reviewing my homework problems and I can't seem to understand the logic behind these correct answers.
The matrix <[1, 0], [0, 0]> has a trace of 1, is 2x2, and uses real entries while having a zero vector. Why is the answer 'does not contain a zero vector'?
Does the zero vector mean zero's in the entire matrix?
I was under the impression that a matrix can be broken up into n vectors so [1,0] is one vector and [0,0] is another vector thereby meaning that there is a zero vector.
linear-algebra vector-spaces vectors matrix-calculus
asked 1 hour ago
S. Snake
163
New contributor
S. Snake is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
4
You have to be careful about your definition of vector. If your vector space is the space of two by two matrices, then the vectors are two by two matrices.
– John Douma
1 hour ago
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
The trace of a square n×n matrix A=(aij) is the sum a11+a22+⋯+ann of the entries on its main diagonal.
Let V be the vector space of all 2×2 matrices with real entries. Let H be the set of all 2×2 matrices with real entries that have trace 1. Is H a subspace of the vector space V?
Does H contain the zero vector of V?
H does not contain the zero vector of V
Hello, I was reviewing my homework problems and I can't seem to understand the logic behind these correct answers.
The matrix <[1, 0], [0, 0]> has a trace of 1, is 2x2, and uses real entries while having a zero vector. Why is the answer 'does not contain a zero vector'?
Does the zero vector mean zero's in the entire matrix?
I was under the impression that a matrix can be broken up into n vectors so [1,0] is one vector and [0,0] is another vector thereby meaning that there is a zero vector.
linear-algebra vector-spaces vectors matrix-calculus
asked 1 hour ago
S. Snake
163
New contributor
S. Snake is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
The trace of a square n×n matrix A=(aij) is the sum a11+a22+⋯+ann of the entries on its main diagonal.
Let V be the vector space of all 2×2 matrices with real entries. Let H be the set of all 2×2 matrices with real entries that have trace 1. Is H a subspace of the vector space V?
Does H contain the zero vector of V?
H does not contain the zero vector of V
Hello, I was reviewing my homework problems and I can't seem to understand the logic behind these correct answers.
The matrix <[1, 0], [0, 0]> has a trace of 1, is 2x2, and uses real entries while having a zero vector. Why is the answer 'does not contain a zero vector'?
Does the zero vector mean zero's in the entire matrix?
I was under the impression that a matrix can be broken up into n vectors so [1,0] is one vector and [0,0] is another vector thereby meaning that there is a zero vector.
linear-algebra vector-spaces vectors matrix-calculus
linear-algebra vector-spaces vectors matrix-calculus
asked 1 hour ago
S. Snake
163
New contributor
S. Snake is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 1 hour ago
S. Snake
163
New contributor
S. Snake is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 1 hour ago
S. Snake
163
New contributor
S. Snake is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 1 hour ago
S. Snake
163
asked 1 hour ago
S. Snake
163
163
New contributor
S. Snake is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
S. Snake is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
S. Snake is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
4
You have to be careful about your definition of vector. If your vector space is the space of two by two matrices, then the vectors are two by two matrices.
– John Douma
1 hour ago
add a comment |Â
4
You have to be careful about your definition of vector. If your vector space is the space of two by two matrices, then the vectors are two by two matrices.
– John Douma
1 hour ago
4
4
You have to be careful about your definition of vector. If your vector space is the space of two by two matrices, then the vectors are two by two matrices.
– John Douma
1 hour ago
You have to be careful about your definition of vector. If your vector space is the space of two by two matrices, then the vectors are two by two matrices.
– John Douma
1 hour ago
add a comment |Â
2 Answers
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3
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The zero vector of your vector space is the $2$ by $2$ matrix whose entries are all zeros. The trace of such matrix is zero not one.
Thus $H$ is not a subspace.
Note that $H$ is not closed under addition or scalar multiplication because the trace is not preserved under these oprations.
answered 56 mins ago
Mohammad Riazi-Kermani
37.7k41957
add a comment |Â
up vote
1
down vote
Notice the difference between the
"geometrical notion" of vector, which is a tuple with numbers in it, for example $(2,3) $ and the
"algebra notion" of vector, which is any element $v $ in a vector space $V $.
In this case, you vector space is filled with matrices, so they are your vectors. Of course one can think of a matrix as being composed of vectors, in the "geometrical notion".
In the algebra side of things, the "zero vector" is the element which does nothing when added to other elements, which is of course the matrix that only has 0s.
answered 50 mins ago
RGS
8,78111129
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
The zero vector of your vector space is the $2$ by $2$ matrix whose entries are all zeros. The trace of such matrix is zero not one.
Thus $H$ is not a subspace.
Note that $H$ is not closed under addition or scalar multiplication because the trace is not preserved under these oprations.
answered 56 mins ago
Mohammad Riazi-Kermani
37.7k41957
add a comment |Â
up vote
3
down vote
The zero vector of your vector space is the $2$ by $2$ matrix whose entries are all zeros. The trace of such matrix is zero not one.
Thus $H$ is not a subspace.
Note that $H$ is not closed under addition or scalar multiplication because the trace is not preserved under these oprations.
answered 56 mins ago
Mohammad Riazi-Kermani
37.7k41957
add a comment |Â
up vote
3
down vote
up vote
3
down vote
The zero vector of your vector space is the $2$ by $2$ matrix whose entries are all zeros. The trace of such matrix is zero not one.
Thus $H$ is not a subspace.
Note that $H$ is not closed under addition or scalar multiplication because the trace is not preserved under these oprations.
answered 56 mins ago
Mohammad Riazi-Kermani
37.7k41957
The zero vector of your vector space is the $2$ by $2$ matrix whose entries are all zeros. The trace of such matrix is zero not one.
Thus $H$ is not a subspace.
Note that $H$ is not closed under addition or scalar multiplication because the trace is not preserved under these oprations.
answered 56 mins ago
Mohammad Riazi-Kermani
37.7k41957
answered 56 mins ago
Mohammad Riazi-Kermani
37.7k41957
answered 56 mins ago
Mohammad Riazi-Kermani
37.7k41957
answered 56 mins ago
Mohammad Riazi-Kermani
37.7k41957
37.7k41957
add a comment |Â
add a comment |Â
up vote
1
down vote
Notice the difference between the
"geometrical notion" of vector, which is a tuple with numbers in it, for example $(2,3) $ and the
"algebra notion" of vector, which is any element $v $ in a vector space $V $.
In this case, you vector space is filled with matrices, so they are your vectors. Of course one can think of a matrix as being composed of vectors, in the "geometrical notion".
In the algebra side of things, the "zero vector" is the element which does nothing when added to other elements, which is of course the matrix that only has 0s.
answered 50 mins ago
RGS
8,78111129
add a comment |Â
up vote
1
down vote
Notice the difference between the
"geometrical notion" of vector, which is a tuple with numbers in it, for example $(2,3) $ and the
"algebra notion" of vector, which is any element $v $ in a vector space $V $.
In this case, you vector space is filled with matrices, so they are your vectors. Of course one can think of a matrix as being composed of vectors, in the "geometrical notion".
In the algebra side of things, the "zero vector" is the element which does nothing when added to other elements, which is of course the matrix that only has 0s.
answered 50 mins ago
RGS
8,78111129
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Notice the difference between the
"geometrical notion" of vector, which is a tuple with numbers in it, for example $(2,3) $ and the
"algebra notion" of vector, which is any element $v $ in a vector space $V $.
In this case, you vector space is filled with matrices, so they are your vectors. Of course one can think of a matrix as being composed of vectors, in the "geometrical notion".
In the algebra side of things, the "zero vector" is the element which does nothing when added to other elements, which is of course the matrix that only has 0s.
answered 50 mins ago
RGS
8,78111129
Notice the difference between the
"geometrical notion" of vector, which is a tuple with numbers in it, for example $(2,3) $ and the
"algebra notion" of vector, which is any element $v $ in a vector space $V $.
In this case, you vector space is filled with matrices, so they are your vectors. Of course one can think of a matrix as being composed of vectors, in the "geometrical notion".
In the algebra side of things, the "zero vector" is the element which does nothing when added to other elements, which is of course the matrix that only has 0s.
answered 50 mins ago
RGS
8,78111129
answered 50 mins ago
RGS
8,78111129
answered 50 mins ago
RGS
8,78111129
answered 50 mins ago
RGS
8,78111129
8,78111129
add a comment |Â
add a comment |Â
S. Snake is a new contributor. Be nice, and check out our Code of Conduct.
S. Snake is a new contributor. Be nice, and check out our Code of Conduct.
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4
You have to be careful about your definition of vector. If your vector space is the space of two by two matrices, then the vectors are two by two matrices.
– John Douma
1 hour ago