Direct product of two irreducible representations of a finite group can be decomposed into a direct sum of irreducible representations. So, starting from a single faithful irreducible representation, is it possible generate every other irreducible representation by successively taking direct...
I am trying to find nonzero prime ideals of \mathbb{Z} \oplus \mathbb {Z}, specifically those which are not also maximal.
If I try to do direct sums of prime ideals, the resulting set is not a prime ideal. (e.g., 2 \mathbb{Z} \oplus 3 \mathbb{Z} is not prime since (3,3) \cdot (2,2) = (6,6)\in...
Homework Statement
Hi, everybody!
I'd like to ask you about the direct sum of subspaces...
I refer to two linear algebra books; 1)Friedberg's book, 2)Hoffman's book.
First of all, I write two definitions of direct sum of subspaces...
in the book 1),
Def.1). Let...
Homework Statement
In R^4 which of the following sums U+V, U+W and V+W are direct? Give reasons
And which of these sums equal R^4?
Homework Equations
U = {(0, a, b, a-b) : a,b ∈ R}
V = {(x, y, z, w) : x=y, z=w}
W = {(x, y, z, w) : x=y}
The Attempt at a Solution
I put that none are direct...
This is a basic question in angular momentum in quantum mechanics that I am studying.
I know that \frac{1}{2}\otimes \frac{1}{2} = 1\oplus 0 What would be a strategy to proving the general statement for spin representations j\otimes s =\bigoplus_{l=|s-j|}^{|s+j|} l
I'm having difficulty understanding this concept of uniqueness. What's the precise definition of it? Let say we have some direct sum decomposition,
(1)Are \left( {\begin{array}{*{20}{c}}
{{R_1}} & 0 \\
0 & {{R_2}} \\
\end{array}} \right) and \left( {\begin{array}{*{20}{c}}...
Hi ,
I was trying to understand why or where would the problem arise in the definition of the direct sum for the coproduct/direct sum for the set (1, 1, 1, ...) infinite number of times. I was trying to reason out as follows. I posted this on the set theory but I am not sure how to tag it to...
Hi ,
This is not a homework problem as I have long passed out of college. I was trying to understand why or where would the problem arise in the definition of the direct sum for the coproduct/direct sum for the set (1, 1, 1, ...) infinite number of times. I was trying to reason out as follows...
Homework Statement
show that if V=M \oplus N, then V^*=M^o+N^o
2. The attempt at a solution
So I need to prove for any f \in V*, f(\epsilon)=(g+h)(\epsilon), where g\in M^o and h\in N^o.
(g+h)(\epsilon)=g(\epsilon)+h(\epsilon)=g(\alpha+\beta)+h(\alpha+\beta)=g(\beta)+h(\alpha), where\alpha...
I know that if V is a direct sum of U and W,
then
1. V=U+W
2 there is no intersection between U and W
However, in some books there is an equivalent condition:
3.Every v can be expressed uniquely as u+w
Why's that? Why can we be so sure about the word "unique"? Thanks.
Show that the direct sum of 2 nonzero rings is never an integral domain
I started by thinking about what a direct sum is
(a,b)(c,d)=(ac,bd)
(a,b)+(c,d)=(a+c,b+d)
We have an integral domain if ab=0 implies a=0 or b=0
For a commutative ring R and an ideal I, is it true that I \oplus R/I \cong R ? I know in some cases this is true, and I know it's true for finitely-generated Abelian groups, but is it true for any commutative ring?
In other words, we know that R/I is isomorphic to some ideal in R, call this...
Homework Statement
Let W1 and W2 be subspaces of a vector space V. Prove that W_1\oplus{}W_2=V \iff each vector in V can be uniquely written as x1+x2=v, where x_1\in W_1 and x_2\in W_2Homework Equations
W_1\oplus{}W_2=V means W_1\cap W_2 =\{0\}, W_1 + W_2 =V and W1 & W2 are subspaces of V
8...
For a direct sum of Mobius band, it is trivial if it has two linear independent nonvanishing sections. I have the following as my sections：
s1=(E^(i*theta), (Cos(theta/2), Sin(theta/2))
s2=(E^(i*theta), (-Sin(theta/2), Cos(theta/2))
Clearly, the above sections are linearly independent and...
Homework Statement
[PLAIN]http://img571.imageshack.us/img571/1821/subspaces.png
Homework Equations
The Attempt at a Solution
Is my solution correct?:
For a,b\in \mathbb{C}
let A=\begin{bmatrix} a \\ a \\ 0 \end{bmatrix}\in U and B=\begin{bmatrix} 0 \\ b \\ b...
I was sad to find out that if H is a normal subgroup of G, we can't say G \cong H \oplus G/H. Now I'm wondering: in which cases does this equality hold?
Homework Statement
17. Let U = f(x; y; 0) : x 2 R; y 2 Rg, E1 = f(x; 0; 0) : x 2 Rg, and E3 = f(0; 0; x) :
x 2 Rg: Are the following assertions true or false? Explain.
(a) U + E1 is a subspace of R3:
(b) U E1 is a direct sum decomposition of U + E1:
(c) U E3 is a direct sum...
Dear all,
I'm reading the tensor part of "A course in modern mathematical physics" by Szekeres and I have trouble understanding a concept that you can find in the attached image of the book page. What are the elements of F(V)? If my understanding of (external) direct sums of vector spaces is...
Let U and V be subspaces of a vector space W. If W=U \oplus V, show U \bigcap V={0}.
I'm a bit lost on this one... as I thought this was essentially the definition of direct sum. I'm unsure where to start. Any help would be great!
This may be a dumb question, but I just want to make sure I understand this correctly.
For R_{1}, R_{2}, ..., R_{n}
R_{1} \oplus R_{2} \oplus, ..., R_{n}=(a_{1},a_{2},...,a_{n})|a_{i} \in R_{i}
does this mean that a ring which is a direct sum of other rings is composed of specific elements...
Homework Statement
Here's the question... it was easier to format it in paint haha:
Please note I'll just write + to mean the plus with the circle around it (direct sum). + is just a normal addition.
Homework Equations
The Attempt at a Solution
V = im(T) + ker(S) means that...
Hi,
I am edging my way towards Dolbeault cohomology on a complex manifold and one of the constructions involves taking the kth exterior product of a direct sum (the decomposition of the cotangent bundle into holomorphic and antiholomorphic subspaces). This relies on a theorem from multilinear...
What's the difference (if any) between a direct sum and a direct product of rings?
For example, in Ireland and Rosen's number theory text, they mention that, in the context of rings, the Chinese Remainder Theorem implies that \mathbb{Z}/(m_1 \cdots...
I'm going through Axler's book and just got introduced the concept of sums of subspaces and the direct sums.
Here's one of the examples he has.
Now the other examples he had were kind of trivial (such as \mathbb{R}^2 = U \oplus W where U = \{ (x,0) | x \in \mathbb{R} \} and W = \{(0,y) |...
Hi everybody,
I'm new to absract algebra and I really can not understand different between direct sum and direct product in group theory (specially abelian groups).
could does anyone give me a clear example or ... ?
thanks
Homework Statement
Let k be a field, V = Mat2x2(k), U:={[a, b], [-b, a] a, b E k} and W:={[a, b], [b, -a] a, b E k}. Show that V is the direct sum of U and W.
Homework Equations
The Attempt at a Solution
Add the matrix for U to the matrix of W. Values in that matrix still exist...
Let V = R^3. Let W be the space generated by w = (1, 0, 0), and let U be the subspace generated by u_1 = (1, 1, 0) and u_2 = (0, 1, 1). Show that V is the direct sum of W and U.
I wasn't sure where to put this since this is under group theory. I am having a little bit trouble understand when to use direct product vs direct sum. One question I have about this is is that if you have two vector spaces that are orthogonal to each other (and example of this might be two...
Homework Statement
Let Pn denote the vector space of polynomials of degree less than or equal to n, and of the form p(x)=p0+p1x+...+pnxn, where the coefficients pi are all real. Let PE denote the subspace of all even polynomials in Pn, i.e., those that satisfy the property p(-x)=p(x)...
Homework Statement
Find a subgroup of Z_4 \oplus Z_2 that is not of the form H \oplus K where H is a subgroup of Z_4 and K is a subgroup of Z_2.
The attempt at a solution
I'm guessing I need to find an H \oplus K where either H or K is not a subgroup. But this seems impossible. Obviously...
We are asked to show that L(SO(4)) = L(SU(2)) (+) L(SU(2))
where L is the Lie algebra and (+) is the direct product.
We are given the hint to consider the antisymmetric 4 by 4 matrices where each row and column has a single 1/2 or -1/2 in it.
By doing this we generate four matrices...
V is a subspace of R^4
V={(x, -y, 2x+y, x-2y): x,y E R}
1) extend {(2,-1,5,0)} to a basis of V.
2) find subspace W of R^4 for which R^4= direct sum V(+)W.
solution:
1)the dimension of V is 2.therefore i need to add one more vector to (2,-1,5,0).
the 2nd vector is (1,0,2,1)...
Okey, I have some silly problems with simple definitions.
The usual sum, which I know, of two vector spaces is a set which consists of all sums of the vectors. A+B=\{a+b|a\in A,\; b\in B\}. Is this the same thing as the direct sum?
I think I saw somewhere (I don't remeber where) a definition...
"If V is finite dimensional, and W is a subspace of V, prove that if T(W)\subset W, there's always a W' such that W' (direct sum) W = V and T(W')\subset W'."
I can't find such a W'.
"Suppose that T : V -> V is a linear transformation of vector spaces over
R whose minimal polynomial has no multiple roots. Show that V can be
expressed as a direct sum
V = V1 + V2 + · · · + Vt
of T-stable subspaces of dimensions at most 2. Show that, relative to a suitable basis, T...
If I have an abelian group A that is the direct sum of cyclic groups, say
A=[tex]C_5 \oplus C_35[\tex], would I be right in saying the annihilator of A (viewed as a Z-module) is generated by (5,35)? If not, how do I find it?