Why is the tensor product of two vector spaces a vector space?
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We defined the tensor product of $v$ and $w$ to just mean a "symbol" $e_{vw}$, then considered the subspace spanned by all these symbols, and finally we quotient out by relations to make the tensor product between two vectors bilinear. The resulting vector space is the tensor product of $V$ and $W$.
I don't understand why this is a vector space.. sure, we've said it's the span of a bunch of symbols, but what does it even mean to "add" together two symbols? How is equality defined in our vector space?
vector-spaces tensor-products tensors
asked 2 days ago
Saad
500210
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We defined the tensor product of $v$ and $w$ to just mean a "symbol" $e_{vw}$, then considered the subspace spanned by all these symbols, and finally we quotient out by relations to make the tensor product between two vectors bilinear. The resulting vector space is the tensor product of $V$ and $W$.
I don't understand why this is a vector space.. sure, we've said it's the span of a bunch of symbols, but what does it even mean to "add" together two symbols? How is equality defined in our vector space?
vector-spaces tensor-products tensors
asked 2 days ago
Saad
500210
2
You can "generate" the smallest vector space containing your "symbols" $e_{vw}$, no problem. It will be the linear span, as you said. A linear span is a vector space by definition.
– Dietrich Burde
2 days ago
If $V$ is $n$-dimensional, and $W$ is $m$-dimensional, then the tensor product is $nm$-dimensional. So as a vector space, $V otimes W$ is just $Bbb{R}^{nm}$. The "symbols" $e_{vw}$ just correspond to the standard basis vectors in $Bbb{R}^{nm}$, after choosing some ordering of the bases of $V$ and $W$.
– Nick
2 days ago
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
We defined the tensor product of $v$ and $w$ to just mean a "symbol" $e_{vw}$, then considered the subspace spanned by all these symbols, and finally we quotient out by relations to make the tensor product between two vectors bilinear. The resulting vector space is the tensor product of $V$ and $W$.
I don't understand why this is a vector space.. sure, we've said it's the span of a bunch of symbols, but what does it even mean to "add" together two symbols? How is equality defined in our vector space?
vector-spaces tensor-products tensors
asked 2 days ago
Saad
500210
We defined the tensor product of $v$ and $w$ to just mean a "symbol" $e_{vw}$, then considered the subspace spanned by all these symbols, and finally we quotient out by relations to make the tensor product between two vectors bilinear. The resulting vector space is the tensor product of $V$ and $W$.
I don't understand why this is a vector space.. sure, we've said it's the span of a bunch of symbols, but what does it even mean to "add" together two symbols? How is equality defined in our vector space?
vector-spaces tensor-products tensors
vector-spaces tensor-products tensors
asked 2 days ago
Saad
500210
asked 2 days ago
Saad
500210
asked 2 days ago
Saad
500210
asked 2 days ago
Saad
500210
asked 2 days ago
Saad
500210
500210
2
You can "generate" the smallest vector space containing your "symbols" $e_{vw}$, no problem. It will be the linear span, as you said. A linear span is a vector space by definition.
– Dietrich Burde
2 days ago
If $V$ is $n$-dimensional, and $W$ is $m$-dimensional, then the tensor product is $nm$-dimensional. So as a vector space, $V otimes W$ is just $Bbb{R}^{nm}$. The "symbols" $e_{vw}$ just correspond to the standard basis vectors in $Bbb{R}^{nm}$, after choosing some ordering of the bases of $V$ and $W$.
– Nick
2 days ago
add a comment |
2
You can "generate" the smallest vector space containing your "symbols" $e_{vw}$, no problem. It will be the linear span, as you said. A linear span is a vector space by definition.
– Dietrich Burde
2 days ago
If $V$ is $n$-dimensional, and $W$ is $m$-dimensional, then the tensor product is $nm$-dimensional. So as a vector space, $V otimes W$ is just $Bbb{R}^{nm}$. The "symbols" $e_{vw}$ just correspond to the standard basis vectors in $Bbb{R}^{nm}$, after choosing some ordering of the bases of $V$ and $W$.
– Nick
2 days ago
2
2
You can "generate" the smallest vector space containing your "symbols" $e_{vw}$, no problem. It will be the linear span, as you said. A linear span is a vector space by definition.
– Dietrich Burde
2 days ago
You can "generate" the smallest vector space containing your "symbols" $e_{vw}$, no problem. It will be the linear span, as you said. A linear span is a vector space by definition.
– Dietrich Burde
2 days ago
If $V$ is $n$-dimensional, and $W$ is $m$-dimensional, then the tensor product is $nm$-dimensional. So as a vector space, $V otimes W$ is just $Bbb{R}^{nm}$. The "symbols" $e_{vw}$ just correspond to the standard basis vectors in $Bbb{R}^{nm}$, after choosing some ordering of the bases of $V$ and $W$.
– Nick
2 days ago
If $V$ is $n$-dimensional, and $W$ is $m$-dimensional, then the tensor product is $nm$-dimensional. So as a vector space, $V otimes W$ is just $Bbb{R}^{nm}$. The "symbols" $e_{vw}$ just correspond to the standard basis vectors in $Bbb{R}^{nm}$, after choosing some ordering of the bases of $V$ and $W$.
– Nick
2 days ago
add a comment |
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You can "generate" the smallest vector space containing your "symbols" $e_{vw}$, no problem. It will be the linear span, as you said. A linear span is a vector space by definition.
– Dietrich Burde
2 days ago
If $V$ is $n$-dimensional, and $W$ is $m$-dimensional, then the tensor product is $nm$-dimensional. So as a vector space, $V otimes W$ is just $Bbb{R}^{nm}$. The "symbols" $e_{vw}$ just correspond to the standard basis vectors in $Bbb{R}^{nm}$, after choosing some ordering of the bases of $V$ and $W$.
– Nick
2 days ago