Matrices as Outer Direct Sum of Vector Spaces
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So studying linear algebra I encountered the outer direct sum and the direct product of collections of vector spaces (which are the same if the collections are finite). When thinking about the uses of these operations (within the reals of linear algebra, that is), I thought about matrices; so now the question:
Can/Should one think of elements of the outer direct sum as matrices?
And if so, could one then generalize, through this thinking, the notion of matrices to infinite matrices?
Thanks in advance!
linear-algebra direct-sum direct-product
asked Nov 17 at 2:27
BlondCafé
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0
down vote
favorite
So studying linear algebra I encountered the outer direct sum and the direct product of collections of vector spaces (which are the same if the collections are finite). When thinking about the uses of these operations (within the reals of linear algebra, that is), I thought about matrices; so now the question:
Can/Should one think of elements of the outer direct sum as matrices?
And if so, could one then generalize, through this thinking, the notion of matrices to infinite matrices?
Thanks in advance!
linear-algebra direct-sum direct-product
asked Nov 17 at 2:27
BlondCafé
71
If $V$ and $W$ are separate vector spaces, you should think of elements of the outer direct sum $V oplus W$ as being pairs $(v, w)$ with the scalar action $lambda (v, w) = (lambda v, lambda w)$. It can also be useful to consider $(v, w)$ as a column vector, since this is compatible with block matrices. But I wouldn’t think of an element of $V oplus W$ as a matrix.
– Joppy
Nov 17 at 7:41
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
So studying linear algebra I encountered the outer direct sum and the direct product of collections of vector spaces (which are the same if the collections are finite). When thinking about the uses of these operations (within the reals of linear algebra, that is), I thought about matrices; so now the question:
Can/Should one think of elements of the outer direct sum as matrices?
And if so, could one then generalize, through this thinking, the notion of matrices to infinite matrices?
Thanks in advance!
linear-algebra direct-sum direct-product
asked Nov 17 at 2:27
BlondCafé
71
So studying linear algebra I encountered the outer direct sum and the direct product of collections of vector spaces (which are the same if the collections are finite). When thinking about the uses of these operations (within the reals of linear algebra, that is), I thought about matrices; so now the question:
Can/Should one think of elements of the outer direct sum as matrices?
And if so, could one then generalize, through this thinking, the notion of matrices to infinite matrices?
Thanks in advance!
linear-algebra direct-sum direct-product
linear-algebra direct-sum direct-product
asked Nov 17 at 2:27
BlondCafé
71
asked Nov 17 at 2:27
BlondCafé
71
asked Nov 17 at 2:27
BlondCafé
71
asked Nov 17 at 2:27
BlondCafé
71
asked Nov 17 at 2:27
BlondCafé
71
71
If $V$ and $W$ are separate vector spaces, you should think of elements of the outer direct sum $V oplus W$ as being pairs $(v, w)$ with the scalar action $lambda (v, w) = (lambda v, lambda w)$. It can also be useful to consider $(v, w)$ as a column vector, since this is compatible with block matrices. But I wouldn’t think of an element of $V oplus W$ as a matrix.
– Joppy
Nov 17 at 7:41
add a comment |
If $V$ and $W$ are separate vector spaces, you should think of elements of the outer direct sum $V oplus W$ as being pairs $(v, w)$ with the scalar action $lambda (v, w) = (lambda v, lambda w)$. It can also be useful to consider $(v, w)$ as a column vector, since this is compatible with block matrices. But I wouldn’t think of an element of $V oplus W$ as a matrix.
– Joppy
Nov 17 at 7:41
If $V$ and $W$ are separate vector spaces, you should think of elements of the outer direct sum $V oplus W$ as being pairs $(v, w)$ with the scalar action $lambda (v, w) = (lambda v, lambda w)$. It can also be useful to consider $(v, w)$ as a column vector, since this is compatible with block matrices. But I wouldn’t think of an element of $V oplus W$ as a matrix.
– Joppy
Nov 17 at 7:41
If $V$ and $W$ are separate vector spaces, you should think of elements of the outer direct sum $V oplus W$ as being pairs $(v, w)$ with the scalar action $lambda (v, w) = (lambda v, lambda w)$. It can also be useful to consider $(v, w)$ as a column vector, since this is compatible with block matrices. But I wouldn’t think of an element of $V oplus W$ as a matrix.
– Joppy
Nov 17 at 7:41
add a comment |
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If $V$ and $W$ are separate vector spaces, you should think of elements of the outer direct sum $V oplus W$ as being pairs $(v, w)$ with the scalar action $lambda (v, w) = (lambda v, lambda w)$. It can also be useful to consider $(v, w)$ as a column vector, since this is compatible with block matrices. But I wouldn’t think of an element of $V oplus W$ as a matrix.
– Joppy
Nov 17 at 7:41