Understanding the concept of direct sum

I suspect that the picture of three axes in R^3 will be your counterexample. In summary, the direct sum of subspaces U1, U2, and U3 exists if and only if it satisfies the condition that a target space Y can be extended uniquely to a map X --> Y, and the union of the three subspaces generate X, and they intersect pairwise in zero. However, the converse may not hold and the sum of any two subspaces may meet the third in zero.
  • #1
JD_PM
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TL;DR Summary
I want to understand what I am missing in the generalization I made for the direct sum when ##3## subspaces are involved.
Given two subspaces ##U_1, U_2##, I understand the concept of direct sum

$$ W= U_1 \oplus U_2 \iff W= U_1 + U_2, \quad U_1 \cap U_2 = \{ 0 \}$$

Where ##W## is a subspace of ##V##.

I am trying to generalize it for more than ##2## subspaces, say ##3##. I thought of the following.

$$ W= U_1 \oplus U_2 \oplus U_3 \iff U_1 \cap U_2 = \{ 0 \}, U_1 \cap U_3 = \{ 0 \}, U_2 \cap U_3 = \{ 0 \}, U_1 + U_2 + U_3 = W $$

It does not seem to have the same structure that for the statement with ##k## subspaces

\begin{align*}
W= U_1 \oplus U_2 \oplus ... \oplus U_k \iff& U_i \cap \left(U_1 + ... + U_{i-1} + U_{i+1} + ... + U_k\right) = \{ 0 \} \\
&U_1 + U_2 + ... + U_k = W
\end{align*}

In particular, the issue lies on the intersection statement. Might you explain why my thought is faulty? I should be able to find a counterexample once I see it :)

Thanks! :biggrin:
 
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  • #2
Let ##W=\mathbb{R}^2##, ##U_1=span( (1,0)),U_2=span(1,1),U_3=span(0,1)## be three one dimensional subspaces Under your attempt, we would have ##W=U_1\oplus U_2 \oplus U_3## but that's not desirable for the definition of direct sum.
 
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  • #3
Office_Shredder said:
Let ##W=\mathbb{R}^2##, ##U_1=span( (1,0)),U_2=span(1,1),U_3=span(0,1)##

But, if I am not mistaken, given

$$U_1 = \{ (x, 0) | x \in \Bbb R\}, \quad U_2 = \{ (x, y) | x, y \in \Bbb R\}, \quad U_3 = \{ (0, y) | y \in \Bbb R\}$$

We could find ##\Bbb R^2= U_1 + U_2 + U_3## in more than one way. Another would be say ##U_1=span( (1,0)),U_2=span(1,2),U_3=span(0,1)##. So I do not see it as a counterexample as the sum is not unique.

Am I missing something?EDIT: Oops I realized you meant

$$U_1 = \{ (x, 0) | x \in \Bbb R\}, \quad U_2 = \{ (y, y) | y \in \Bbb R\}, \quad U_3 = \{ (0, y) | y \in \Bbb R\}$$

So indeed, we can write ##\Bbb R^2= U_1 + U_2 + U_3## in a unique way (up to a scalar factor). EDIT 2: " we can write ##\Bbb R^2= U_1 + U_2 + U_3## in a unique way (up to a scalar factor)" is not true, see #5.
 
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  • #4
I have been thinking that the key might be in finding a counterexample that satisfies ##W= U_1 \oplus U_2 \oplus U_3## but fails to meet ##\left(U_1+U_2\right)\cap U_3 = \{ 0 \}##.

However I do not seem to see it... Might you give me a hint?
 
  • #5
Let's start all over again.

I found a more convincing definition of direct sum (Axler, page 21)

##U_1 + U_2 + \ ... \ + U_k## is a direct sum if ##x \in U_1 + U_2 + \ ... \ + U_k## can be written in a unique way as ##x = u_1 + u_2 + \ ... \ + u_k##, where ##u_i \in U_i##

JD_PM said:
$$U_1 = \{ (x, 0) | x \in \Bbb R\}, \quad U_2 = \{ (y, y) | y \in \Bbb R\}, \quad U_3 = \{ (0, y) | y \in \Bbb R\}$$

##\Bbb R^2= U_1 \oplus U_2 \oplus U_3## does not hold in this case because there are (at least) two ways of obtaining the zero vector

$$(0, 0) = (1, 0) + (-1, -1) + (0, 1)$$

And the trivial case

$$(0, 0) = (0, 0) + (0, 0) + (0, 0)$$

So, if I am not mistaken, that is the reason why that is a valid counterexample to the given definition of the direct sum :smile:
 
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  • #6
There are several equivalent formulations in finite dimensions. Equivalent are:
  1. ##\sum X_i ## is a direct sum
  2. ## 0 ## decomposes uniquely
  3. ##X_j\cap \sum _{i\neq j} X_i = \{0\} ## for every ##i=1,\ldots, n##
  4. ##\dim \sum X_i = \sum \dim X_i ##
 
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  • #7
JD_PM said:
So, if I am not mistaken, that is the reason why that is a valid counterexample to the given definition of the direct sum :smile:
You got it, but allow me to be a little nitpicky about your words. It is a counterexample to your definition attempt being a correct definition of the direct sum.
 
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  • #8
The first step is to master the correct definition of a direct sum, from category theory. A direct sum of three spaces A,B,C is a space X with 3 subspaces A,B,C such that for any target space Y, any three maps A-->Y, B-->Y, C-->Y, always extends uniquely to a map X-->Y. It follows immediately that the union of the three spaces generate X, and that they intersect pairwise in zero. You, may then try to check if the converse holds. I doubt it, by visualizing three lines in the plane, all through zero. Youm might want to add that the sum of any two meets the third in zero.
 

What is a direct sum?

A direct sum is a mathematical concept that involves combining two or more objects in a way that preserves their individual identities. In other words, it is a way of combining objects without losing any information about their individual properties.

How is a direct sum different from a direct product?

While a direct sum involves combining objects in a way that preserves their individual identities, a direct product involves combining objects in a way that creates a new object with properties that are a combination of the individual objects.

What are some real-world examples of direct sums?

One common example of a direct sum is combining two vectors in physics. Another example is combining two sets of numbers in mathematics.

How is a direct sum represented mathematically?

In mathematics, a direct sum is often represented using the symbol ⊕ or ⊕⊕. This symbol is placed between the objects being combined to indicate that they are being added together in a direct sum.

What is the significance of direct sums in mathematics?

Direct sums are important in mathematics because they allow us to combine objects in a way that preserves their individual properties. This can be useful in solving problems and understanding complex systems.

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