Linear algebra with matrices not only for $mathbbR^n$?
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I haven't taken a course on Linear Algebra, but I kind of used the concepts of row/column vectors, matrices a lot. But somehow only with respect with the $mathbbR^n$ vector space. When talking about other vector spaces, like a function space, do matrices (and implicitly column/row vectors) have that great impact?
My intuition says yes, because:
In $mathbbR^3$:
Let $B = e_1=(1,0,0),e_2=(0,1,0),e_3=(0,0,1)$ be a basis of $mathbbR^3$, where (a,b,c) is a 3-tuple
Then:
$[(1,2,3)]_B = beginbmatrix1\2\3endbmatrix$In the function space of polynomials:
Let $B = 1,x,x^2$ be a basis
Then:
$[1+2x+3x^2]_B = beginbmatrix1\2\3endbmatrix$.
Are these considerations useful or good to be kept in mind when working with linear algebra? I mean, is it important not to forget about matrices when turning from $mathbbR^n$ to another vector space?
linear-algebra matrices vector-spaces
asked Jun 15 at 15:17
user_anon
508
add a comment |Â
up vote
1
down vote
favorite
I haven't taken a course on Linear Algebra, but I kind of used the concepts of row/column vectors, matrices a lot. But somehow only with respect with the $mathbbR^n$ vector space. When talking about other vector spaces, like a function space, do matrices (and implicitly column/row vectors) have that great impact?
My intuition says yes, because:
In $mathbbR^3$:
Let $B = e_1=(1,0,0),e_2=(0,1,0),e_3=(0,0,1)$ be a basis of $mathbbR^3$, where (a,b,c) is a 3-tuple
Then:
$[(1,2,3)]_B = beginbmatrix1\2\3endbmatrix$In the function space of polynomials:
Let $B = 1,x,x^2$ be a basis
Then:
$[1+2x+3x^2]_B = beginbmatrix1\2\3endbmatrix$.
Are these considerations useful or good to be kept in mind when working with linear algebra? I mean, is it important not to forget about matrices when turning from $mathbbR^n$ to another vector space?
linear-algebra matrices vector-spaces
asked Jun 15 at 15:17
user_anon
508
2
Yes, it is very important. As you note, using bases and hence coordinate vectors in the general vector space, a lot of questions can be answered using the theory developed for $R^n $ or even $n $-tuples over a generic field, and matrices over that field
– AnyAD
Jun 15 at 15:22
6
Matrices can be used for any finite dimensional vector space over any field ($mathbf Q, mathbf C$ for instance), and even for any finitely generated free module over any commutative ring (modules are the structure corresponding to vector spaces when working over a ring instead of a field).
– Bernard
Jun 15 at 15:22
@Bernard Curiosity: you can actually drop the requirement that $R$ is commutative, but the opposite ring $R^rm op$ starts popping up. There are a couple of results in this direction in Adkins & Weintraub's Algebra: An Approach via Module Theory, if I recall correctly.
– Ivo Terek
Jun 15 at 15:31
Also, that importance to matrices is given because there is the concept of basis, right?
– user_anon
Jun 15 at 15:36
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I haven't taken a course on Linear Algebra, but I kind of used the concepts of row/column vectors, matrices a lot. But somehow only with respect with the $mathbbR^n$ vector space. When talking about other vector spaces, like a function space, do matrices (and implicitly column/row vectors) have that great impact?
My intuition says yes, because:
In $mathbbR^3$:
Let $B = e_1=(1,0,0),e_2=(0,1,0),e_3=(0,0,1)$ be a basis of $mathbbR^3$, where (a,b,c) is a 3-tuple
Then:
$[(1,2,3)]_B = beginbmatrix1\2\3endbmatrix$In the function space of polynomials:
Let $B = 1,x,x^2$ be a basis
Then:
$[1+2x+3x^2]_B = beginbmatrix1\2\3endbmatrix$.
Are these considerations useful or good to be kept in mind when working with linear algebra? I mean, is it important not to forget about matrices when turning from $mathbbR^n$ to another vector space?
linear-algebra matrices vector-spaces
asked Jun 15 at 15:17
user_anon
508
I haven't taken a course on Linear Algebra, but I kind of used the concepts of row/column vectors, matrices a lot. But somehow only with respect with the $mathbbR^n$ vector space. When talking about other vector spaces, like a function space, do matrices (and implicitly column/row vectors) have that great impact?
My intuition says yes, because:
In $mathbbR^3$:
Let $B = e_1=(1,0,0),e_2=(0,1,0),e_3=(0,0,1)$ be a basis of $mathbbR^3$, where (a,b,c) is a 3-tuple
Then:
$[(1,2,3)]_B = beginbmatrix1\2\3endbmatrix$In the function space of polynomials:
Let $B = 1,x,x^2$ be a basis
Then:
$[1+2x+3x^2]_B = beginbmatrix1\2\3endbmatrix$.
Are these considerations useful or good to be kept in mind when working with linear algebra? I mean, is it important not to forget about matrices when turning from $mathbbR^n$ to another vector space?
linear-algebra matrices vector-spaces
asked Jun 15 at 15:17
user_anon
508
edited Aug 13 at 10:24
asked Jun 15 at 15:17
user_anon
508
asked Jun 15 at 15:17
user_anon
508
asked Jun 15 at 15:17
user_anon
508
508
2
Yes, it is very important. As you note, using bases and hence coordinate vectors in the general vector space, a lot of questions can be answered using the theory developed for $R^n $ or even $n $-tuples over a generic field, and matrices over that field
– AnyAD
Jun 15 at 15:22
6
Matrices can be used for any finite dimensional vector space over any field ($mathbf Q, mathbf C$ for instance), and even for any finitely generated free module over any commutative ring (modules are the structure corresponding to vector spaces when working over a ring instead of a field).
– Bernard
Jun 15 at 15:22
@Bernard Curiosity: you can actually drop the requirement that $R$ is commutative, but the opposite ring $R^rm op$ starts popping up. There are a couple of results in this direction in Adkins & Weintraub's Algebra: An Approach via Module Theory, if I recall correctly.
– Ivo Terek
Jun 15 at 15:31
Also, that importance to matrices is given because there is the concept of basis, right?
– user_anon
Jun 15 at 15:36
add a comment |Â
2
Yes, it is very important. As you note, using bases and hence coordinate vectors in the general vector space, a lot of questions can be answered using the theory developed for $R^n $ or even $n $-tuples over a generic field, and matrices over that field
– AnyAD
Jun 15 at 15:22
6
Matrices can be used for any finite dimensional vector space over any field ($mathbf Q, mathbf C$ for instance), and even for any finitely generated free module over any commutative ring (modules are the structure corresponding to vector spaces when working over a ring instead of a field).
– Bernard
Jun 15 at 15:22
@Bernard Curiosity: you can actually drop the requirement that $R$ is commutative, but the opposite ring $R^rm op$ starts popping up. There are a couple of results in this direction in Adkins & Weintraub's Algebra: An Approach via Module Theory, if I recall correctly.
– Ivo Terek
Jun 15 at 15:31
Also, that importance to matrices is given because there is the concept of basis, right?
– user_anon
Jun 15 at 15:36
2
2
Yes, it is very important. As you note, using bases and hence coordinate vectors in the general vector space, a lot of questions can be answered using the theory developed for $R^n $ or even $n $-tuples over a generic field, and matrices over that field
– AnyAD
Jun 15 at 15:22
Yes, it is very important. As you note, using bases and hence coordinate vectors in the general vector space, a lot of questions can be answered using the theory developed for $R^n $ or even $n $-tuples over a generic field, and matrices over that field
– AnyAD
Jun 15 at 15:22
6
6
Matrices can be used for any finite dimensional vector space over any field ($mathbf Q, mathbf C$ for instance), and even for any finitely generated free module over any commutative ring (modules are the structure corresponding to vector spaces when working over a ring instead of a field).
– Bernard
Jun 15 at 15:22
Matrices can be used for any finite dimensional vector space over any field ($mathbf Q, mathbf C$ for instance), and even for any finitely generated free module over any commutative ring (modules are the structure corresponding to vector spaces when working over a ring instead of a field).
– Bernard
Jun 15 at 15:22
@Bernard Curiosity: you can actually drop the requirement that $R$ is commutative, but the opposite ring $R^rm op$ starts popping up. There are a couple of results in this direction in Adkins & Weintraub's Algebra: An Approach via Module Theory, if I recall correctly.
– Ivo Terek
Jun 15 at 15:31
@Bernard Curiosity: you can actually drop the requirement that $R$ is commutative, but the opposite ring $R^rm op$ starts popping up. There are a couple of results in this direction in Adkins & Weintraub's Algebra: An Approach via Module Theory, if I recall correctly.
– Ivo Terek
Jun 15 at 15:31
Also, that importance to matrices is given because there is the concept of basis, right?
– user_anon
Jun 15 at 15:36
Also, that importance to matrices is given because there is the concept of basis, right?
– user_anon
Jun 15 at 15:36
add a comment |Â
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2
Yes, it is very important. As you note, using bases and hence coordinate vectors in the general vector space, a lot of questions can be answered using the theory developed for $R^n $ or even $n $-tuples over a generic field, and matrices over that field
– AnyAD
Jun 15 at 15:22
6
Matrices can be used for any finite dimensional vector space over any field ($mathbf Q, mathbf C$ for instance), and even for any finitely generated free module over any commutative ring (modules are the structure corresponding to vector spaces when working over a ring instead of a field).
– Bernard
Jun 15 at 15:22
@Bernard Curiosity: you can actually drop the requirement that $R$ is commutative, but the opposite ring $R^rm op$ starts popping up. There are a couple of results in this direction in Adkins & Weintraub's Algebra: An Approach via Module Theory, if I recall correctly.
– Ivo Terek
Jun 15 at 15:31
Also, that importance to matrices is given because there is the concept of basis, right?
– user_anon
Jun 15 at 15:36