The direct sum is an operation from abstract algebra, a branch of mathematics. As an example, consider the direct sum , where is the set of real numbers. is the Cartesian plane, the xy-plane from elementary algebra. In general, the direct sum of two objects is another object of the same type, so the direct sum of two geometric objects is a geometric object and the direct sum of two sets is a set.
To see how direct sum is used in abstract algebra, consider a more elementary structure in abstract algebra, the abelian group. The direct sum of two abelian groups and is another abelian group consisting of the ordered pairs where and . To add ordered pairs, we define the sum to be ; in other words addition is defined coordinate-wise. A similar process can be used to form the direct sum of any two algebraic structures, such as rings, modules, and vector spaces.
We can also form direct sums with any number of summands, for example , provided and are the same kinds of algebraic structures, that is, all groups or all rings or all vector spaces.
In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. However, there are other operations which could also be considered as a kind of addition for matrices, the direct sum and the Kronecker sum.
Entrywise sum
Two matrices must have an equal number of rows and columns to be added. The sum of two matrices A and B will be a matrix which has the same number of rows and columns as do A and B. The sum of A and B, denoted A + B, is computed by adding corresponding elements of A and B:
For example:
We can also subtract one matrix from another, as long as they have the same dimensions. A−B is computed by subtracting corresponding elements of A and B, and has the same dimensions as A and B. For example:
Direct sum
Another operation, which is used less often, is the direct sum (denoted by ⊕). Note the Kronecker sum is also denoted ⊕; the context should make the usage clear. The direct sum of any pair of matrices A of size m×n and B of size p×q is a matrix of size (m + p) × (n + q) defined as
In general topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, free sum, topological sum, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology. Roughly speaking, two or more spaces may be considered together, each looking as it would alone.
The name coproduct originates from the fact that the disjoint union is the categorical dual of the product space construction.
Definition
Let {Xi: i ∈ I} be a family of topological spaces indexed by I. Let
be the disjoint union of the underlying sets. For each i in I, let
be the canonical injection (defined by ). The disjoint union topology on X is defined as the finest topology on X for which the canonical injections {φi} are continuous.
Explicitly, the disjoint union topology can be described as follows. A subset U of X is open in Xif and only if its preimage is open in Xi for each i ∈ I.
Oxford Linear Algebra: Direct Sum of Vector Spaces
University of Oxford mathematician Dr Tom Crawford explains the concept of the direct sum of vector subspaces with several fully-worked examples. Check out ProPrep with a 30-day free trial to see how it can help you to improve your performance in STEM-based subjects: https://www.proprep.uk/info/TOM-Crawford
Links to the other videos mentioned:
Subspace Test - https://youtu.be/3_MxBlWQsgs
Dimension Formula: https://youtu.be/lGo9E5dZ4f8
Test your understanding of the content covered in the video with some practice exercises courtesy of ProPrep. You can download the workbooks and solutions for free here: https://api.proprep.com/course/downloadbook?file=Proprep%20-%20Linear%20Algebra%20-%20General%20Vector%20Spaces%20-%20workbook.pdf
You can also find several video lectures from ProPrep e...
published: 26 May 2023
Direct Sum
Have you ever wondered how to sum two mathematical objects in an elegant way? Then this video is for you! In this video, I define the sum of two vector spaces and show something neat: If you add two bases together, you get a basis for the direct sum! Finally, I generalize this notion to direct sums of n vector spaces. Enjoy!
Check out my Vector Space Playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmDhpRvkRXUMNGfV3LVZwL_w
Subscribe to my channel: youtube.com/c/drpeyam
published: 06 May 2019
Linear Algebra 130, Direct Sum
Descrição
published: 21 Mar 2015
Linear Algebra 131, Direct Sum, examples
Descrição
published: 21 Mar 2015
The direct sum and product of vector spaces -- Abstract Linear Algebra 9
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published: 25 Feb 2022
25 - Direct sums of subspaces
Algebra 1M - international
Course no. 104016
Dr. Aviv Censor
Technion - International school of engineering
published: 23 Nov 2015
Direct Sum
Direct Sum definition
In this video, I define the notion of direct sum of n subspaces and show what it has to do with eigenvectors.
Direct sum of two subspaces: https://youtu.be/GjbMddz0Qxs
Check out my Diagonalization playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmCSovHY6cXzPMNSuWOwd9wB
Subscribe to my channel: https://www.youtube.com/c/drpeyam
published: 25 Oct 2019
Direct Sum of Subspaces and its Examples
This lecture explains the direct sum of the subspaces and its illustrative examples.
Other videos @DrHarishGarg
Row reduced Echelon form: https://youtu.be/yCTfbsf60NA
Rank of the matrix: https://youtu.be/Hd34nBwcBG8
Vector Subspaces: https://youtu.be/9uN7CwDqvZQ
Intersection & Union of the subspaces: https://youtu.be/UtUvk1k5YIU
Linear Span and Combination: https://youtu.be/dT0DHyE2Fz4
Direct Sum of Subspace: https://youtu.be/W1RTq9ipXI8
University of Oxford mathematician Dr Tom Crawford explains the concept of the direct sum of vector subspaces with several fully-worked examples. Check out ProP...
University of Oxford mathematician Dr Tom Crawford explains the concept of the direct sum of vector subspaces with several fully-worked examples. Check out ProPrep with a 30-day free trial to see how it can help you to improve your performance in STEM-based subjects: https://www.proprep.uk/info/TOM-Crawford
Links to the other videos mentioned:
Subspace Test - https://youtu.be/3_MxBlWQsgs
Dimension Formula: https://youtu.be/lGo9E5dZ4f8
Test your understanding of the content covered in the video with some practice exercises courtesy of ProPrep. You can download the workbooks and solutions for free here: https://api.proprep.com/course/downloadbook?file=Proprep%20-%20Linear%20Algebra%20-%20General%20Vector%20Spaces%20-%20workbook.pdf
You can also find several video lectures from ProPrep explaining the content covered in the video at the links below.
Direct Sum: https://www.proprep.com/courses/all/linear-algebra/general-vector-spaces/subspaces/vid30219
Sum of Subspaces: https://www.proprep.com/university-of-warwick/warwick-university/ma106/general-vector-spaces/subspaces/vid30220
As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises.
The video begins with the formal mathematical definition of a direct sum via the sum of vector spaces which only have the zero vector in their intersection. This is then shown to be equivalent to an alternative definition which says the representation via the sum of vectors is unique.
Next we look at 3 fully-worked examples where we are asked to check if the given sum of vector spaces is a true direct sum or not. By checking the intersection we see that two of them are not a direct sum as their intersection contains a non-zero vector. For the third example the intersection is shown to be zero, so we then check whether the two subspaces span the larger vector space. By constructing a basis for each subspace, we see that this is indeed the case and thus conclude it is a true direct sum.
Watch the other videos from the Oxford Linear Algebra series at the links below.
Solving Systems of Linear Equations using Elementary Row Operations (ERO’s): https://youtu.be/9pF__coVyEE
Calculating the inverse of 2x2, 3x3 and 4x4 matrices: https://youtu.be/VKOaG3Ogf9Q
What is the Determinant Function: https://www.youtube.com/watch?v=bLsBWVYSg0A
The Easiest Method to Calculate Determinants: https://youtu.be/qniUv4EZB0w
Eigenvalues and Eigenvectors Explained: https://youtu.be/8uISh6xyW7w
Spectral Theorem Proof: https://youtu.be/ADwsk9G5s_8
Vector Space Axioms: https://youtu.be/draqOOUoWQM
Subspace Test: https://youtu.be/3_MxBlWQsgs
Basis, Spanning and Linear Independence: https://youtu.be/wdOFi8aUNp0
Dimension Formula: https://youtu.be/lGo9E5dZ4f8
Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: https://www.seh.ox.ac.uk/people/tom-crawford
For more maths content check out Tom's website https://tomrocksmaths.com/
You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.
https://www.facebook.com/tomrocksmaths/
https://twitter.com/tomrocksmaths
https://www.instagram.com/tomrocksmaths/
Get your Tom Rocks Maths merchandise here:
https://beautifulequations.net/collections/tom-rocks-maths
University of Oxford mathematician Dr Tom Crawford explains the concept of the direct sum of vector subspaces with several fully-worked examples. Check out ProPrep with a 30-day free trial to see how it can help you to improve your performance in STEM-based subjects: https://www.proprep.uk/info/TOM-Crawford
Links to the other videos mentioned:
Subspace Test - https://youtu.be/3_MxBlWQsgs
Dimension Formula: https://youtu.be/lGo9E5dZ4f8
Test your understanding of the content covered in the video with some practice exercises courtesy of ProPrep. You can download the workbooks and solutions for free here: https://api.proprep.com/course/downloadbook?file=Proprep%20-%20Linear%20Algebra%20-%20General%20Vector%20Spaces%20-%20workbook.pdf
You can also find several video lectures from ProPrep explaining the content covered in the video at the links below.
Direct Sum: https://www.proprep.com/courses/all/linear-algebra/general-vector-spaces/subspaces/vid30219
Sum of Subspaces: https://www.proprep.com/university-of-warwick/warwick-university/ma106/general-vector-spaces/subspaces/vid30220
As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises.
The video begins with the formal mathematical definition of a direct sum via the sum of vector spaces which only have the zero vector in their intersection. This is then shown to be equivalent to an alternative definition which says the representation via the sum of vectors is unique.
Next we look at 3 fully-worked examples where we are asked to check if the given sum of vector spaces is a true direct sum or not. By checking the intersection we see that two of them are not a direct sum as their intersection contains a non-zero vector. For the third example the intersection is shown to be zero, so we then check whether the two subspaces span the larger vector space. By constructing a basis for each subspace, we see that this is indeed the case and thus conclude it is a true direct sum.
Watch the other videos from the Oxford Linear Algebra series at the links below.
Solving Systems of Linear Equations using Elementary Row Operations (ERO’s): https://youtu.be/9pF__coVyEE
Calculating the inverse of 2x2, 3x3 and 4x4 matrices: https://youtu.be/VKOaG3Ogf9Q
What is the Determinant Function: https://www.youtube.com/watch?v=bLsBWVYSg0A
The Easiest Method to Calculate Determinants: https://youtu.be/qniUv4EZB0w
Eigenvalues and Eigenvectors Explained: https://youtu.be/8uISh6xyW7w
Spectral Theorem Proof: https://youtu.be/ADwsk9G5s_8
Vector Space Axioms: https://youtu.be/draqOOUoWQM
Subspace Test: https://youtu.be/3_MxBlWQsgs
Basis, Spanning and Linear Independence: https://youtu.be/wdOFi8aUNp0
Dimension Formula: https://youtu.be/lGo9E5dZ4f8
Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: https://www.seh.ox.ac.uk/people/tom-crawford
For more maths content check out Tom's website https://tomrocksmaths.com/
You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.
https://www.facebook.com/tomrocksmaths/
https://twitter.com/tomrocksmaths
https://www.instagram.com/tomrocksmaths/
Get your Tom Rocks Maths merchandise here:
https://beautifulequations.net/collections/tom-rocks-maths
Have you ever wondered how to sum two mathematical objects in an elegant way? Then this video is for you! In this video, I define the sum of two vector spaces a...
Have you ever wondered how to sum two mathematical objects in an elegant way? Then this video is for you! In this video, I define the sum of two vector spaces and show something neat: If you add two bases together, you get a basis for the direct sum! Finally, I generalize this notion to direct sums of n vector spaces. Enjoy!
Check out my Vector Space Playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmDhpRvkRXUMNGfV3LVZwL_w
Subscribe to my channel: youtube.com/c/drpeyam
Have you ever wondered how to sum two mathematical objects in an elegant way? Then this video is for you! In this video, I define the sum of two vector spaces and show something neat: If you add two bases together, you get a basis for the direct sum! Finally, I generalize this notion to direct sums of n vector spaces. Enjoy!
Check out my Vector Space Playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmDhpRvkRXUMNGfV3LVZwL_w
Subscribe to my channel: youtube.com/c/drpeyam
⭐Support the channel⭐
Patreon: https://www.patreon.com/michaelpennmath
Merch: https://teespring.com/stores/michael-penn-math
My amazon shop: https://www.amazon....
⭐Support the channel⭐
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⭐my other channels⭐
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⭐My Links⭐
Personal Website: http://www.michael-penn.net
Instagram: https://www.instagram.com/melp2718/
Randolph College Math: http://www.randolphcollege.edu/mathematics/
Research Gate profile: https://www.researchgate.net/profile/Michael_Penn5
Google Scholar profile: https://scholar.google.com/citations?user=W5wkSxcAAAAJ&hl=en
⭐Support the channel⭐
Patreon: https://www.patreon.com/michaelpennmath
Merch: https://teespring.com/stores/michael-penn-math
My amazon shop: https://www.amazon.com/shop/michaelpenn
⭐my other channels⭐
Main Channel: https://www.youtube.com/michaelpennmath
non-math podcast: https://www.youtube.com/channel/UCIl9oLq0igolmqLYipO0OBQ
⭐My Links⭐
Personal Website: http://www.michael-penn.net
Instagram: https://www.instagram.com/melp2718/
Randolph College Math: http://www.randolphcollege.edu/mathematics/
Research Gate profile: https://www.researchgate.net/profile/Michael_Penn5
Google Scholar profile: https://scholar.google.com/citations?user=W5wkSxcAAAAJ&hl=en
Direct Sum definition
In this video, I define the notion of direct sum of n subspaces and show what it has to do with eigenvectors.
Direct sum of two subspace...
Direct Sum definition
In this video, I define the notion of direct sum of n subspaces and show what it has to do with eigenvectors.
Direct sum of two subspaces: https://youtu.be/GjbMddz0Qxs
Check out my Diagonalization playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmCSovHY6cXzPMNSuWOwd9wB
Subscribe to my channel: https://www.youtube.com/c/drpeyam
Direct Sum definition
In this video, I define the notion of direct sum of n subspaces and show what it has to do with eigenvectors.
Direct sum of two subspaces: https://youtu.be/GjbMddz0Qxs
Check out my Diagonalization playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmCSovHY6cXzPMNSuWOwd9wB
Subscribe to my channel: https://www.youtube.com/c/drpeyam
This lecture explains the direct sum of the subspaces and its illustrative examples.
Other videos @DrHarishGarg
Row reduced Echelon form: https://youtu.be/yCTfb...
This lecture explains the direct sum of the subspaces and its illustrative examples.
Other videos @DrHarishGarg
Row reduced Echelon form: https://youtu.be/yCTfbsf60NA
Rank of the matrix: https://youtu.be/Hd34nBwcBG8
Vector Subspaces: https://youtu.be/9uN7CwDqvZQ
Intersection & Union of the subspaces: https://youtu.be/UtUvk1k5YIU
Linear Span and Combination: https://youtu.be/dT0DHyE2Fz4
Direct Sum of Subspace: https://youtu.be/W1RTq9ipXI8
This lecture explains the direct sum of the subspaces and its illustrative examples.
Other videos @DrHarishGarg
Row reduced Echelon form: https://youtu.be/yCTfbsf60NA
Rank of the matrix: https://youtu.be/Hd34nBwcBG8
Vector Subspaces: https://youtu.be/9uN7CwDqvZQ
Intersection & Union of the subspaces: https://youtu.be/UtUvk1k5YIU
Linear Span and Combination: https://youtu.be/dT0DHyE2Fz4
Direct Sum of Subspace: https://youtu.be/W1RTq9ipXI8
University of Oxford mathematician Dr Tom Crawford explains the concept of the direct sum of vector subspaces with several fully-worked examples. Check out ProPrep with a 30-day free trial to see how it can help you to improve your performance in STEM-based subjects: https://www.proprep.uk/info/TOM-Crawford
Links to the other videos mentioned:
Subspace Test - https://youtu.be/3_MxBlWQsgs
Dimension Formula: https://youtu.be/lGo9E5dZ4f8
Test your understanding of the content covered in the video with some practice exercises courtesy of ProPrep. You can download the workbooks and solutions for free here: https://api.proprep.com/course/downloadbook?file=Proprep%20-%20Linear%20Algebra%20-%20General%20Vector%20Spaces%20-%20workbook.pdf
You can also find several video lectures from ProPrep explaining the content covered in the video at the links below.
Direct Sum: https://www.proprep.com/courses/all/linear-algebra/general-vector-spaces/subspaces/vid30219
Sum of Subspaces: https://www.proprep.com/university-of-warwick/warwick-university/ma106/general-vector-spaces/subspaces/vid30220
As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises.
The video begins with the formal mathematical definition of a direct sum via the sum of vector spaces which only have the zero vector in their intersection. This is then shown to be equivalent to an alternative definition which says the representation via the sum of vectors is unique.
Next we look at 3 fully-worked examples where we are asked to check if the given sum of vector spaces is a true direct sum or not. By checking the intersection we see that two of them are not a direct sum as their intersection contains a non-zero vector. For the third example the intersection is shown to be zero, so we then check whether the two subspaces span the larger vector space. By constructing a basis for each subspace, we see that this is indeed the case and thus conclude it is a true direct sum.
Watch the other videos from the Oxford Linear Algebra series at the links below.
Solving Systems of Linear Equations using Elementary Row Operations (ERO’s): https://youtu.be/9pF__coVyEE
Calculating the inverse of 2x2, 3x3 and 4x4 matrices: https://youtu.be/VKOaG3Ogf9Q
What is the Determinant Function: https://www.youtube.com/watch?v=bLsBWVYSg0A
The Easiest Method to Calculate Determinants: https://youtu.be/qniUv4EZB0w
Eigenvalues and Eigenvectors Explained: https://youtu.be/8uISh6xyW7w
Spectral Theorem Proof: https://youtu.be/ADwsk9G5s_8
Vector Space Axioms: https://youtu.be/draqOOUoWQM
Subspace Test: https://youtu.be/3_MxBlWQsgs
Basis, Spanning and Linear Independence: https://youtu.be/wdOFi8aUNp0
Dimension Formula: https://youtu.be/lGo9E5dZ4f8
Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: https://www.seh.ox.ac.uk/people/tom-crawford
For more maths content check out Tom's website https://tomrocksmaths.com/
You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.
https://www.facebook.com/tomrocksmaths/
https://twitter.com/tomrocksmaths
https://www.instagram.com/tomrocksmaths/
Get your Tom Rocks Maths merchandise here:
https://beautifulequations.net/collections/tom-rocks-maths
Have you ever wondered how to sum two mathematical objects in an elegant way? Then this video is for you! In this video, I define the sum of two vector spaces and show something neat: If you add two bases together, you get a basis for the direct sum! Finally, I generalize this notion to direct sums of n vector spaces. Enjoy!
Check out my Vector Space Playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmDhpRvkRXUMNGfV3LVZwL_w
Subscribe to my channel: youtube.com/c/drpeyam
⭐Support the channel⭐
Patreon: https://www.patreon.com/michaelpennmath
Merch: https://teespring.com/stores/michael-penn-math
My amazon shop: https://www.amazon.com/shop/michaelpenn
⭐my other channels⭐
Main Channel: https://www.youtube.com/michaelpennmath
non-math podcast: https://www.youtube.com/channel/UCIl9oLq0igolmqLYipO0OBQ
⭐My Links⭐
Personal Website: http://www.michael-penn.net
Instagram: https://www.instagram.com/melp2718/
Randolph College Math: http://www.randolphcollege.edu/mathematics/
Research Gate profile: https://www.researchgate.net/profile/Michael_Penn5
Google Scholar profile: https://scholar.google.com/citations?user=W5wkSxcAAAAJ&hl=en
Direct Sum definition
In this video, I define the notion of direct sum of n subspaces and show what it has to do with eigenvectors.
Direct sum of two subspaces: https://youtu.be/GjbMddz0Qxs
Check out my Diagonalization playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmCSovHY6cXzPMNSuWOwd9wB
Subscribe to my channel: https://www.youtube.com/c/drpeyam
This lecture explains the direct sum of the subspaces and its illustrative examples.
Other videos @DrHarishGarg
Row reduced Echelon form: https://youtu.be/yCTfbsf60NA
Rank of the matrix: https://youtu.be/Hd34nBwcBG8
Vector Subspaces: https://youtu.be/9uN7CwDqvZQ
Intersection & Union of the subspaces: https://youtu.be/UtUvk1k5YIU
Linear Span and Combination: https://youtu.be/dT0DHyE2Fz4
Direct Sum of Subspace: https://youtu.be/W1RTq9ipXI8
The direct sum is an operation from abstract algebra, a branch of mathematics. As an example, consider the direct sum , where is the set of real numbers. is the Cartesian plane, the xy-plane from elementary algebra. In general, the direct sum of two objects is another object of the same type, so the direct sum of two geometric objects is a geometric object and the direct sum of two sets is a set.
To see how direct sum is used in abstract algebra, consider a more elementary structure in abstract algebra, the abelian group. The direct sum of two abelian groups and is another abelian group consisting of the ordered pairs where and . To add ordered pairs, we define the sum to be ; in other words addition is defined coordinate-wise. A similar process can be used to form the direct sum of any two algebraic structures, such as rings, modules, and vector spaces.
We can also form direct sums with any number of summands, for example , provided and are the same kinds of algebraic structures, that is, all groups or all rings or all vector spaces.
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