My textbook states that for operators on complex vector spaces with dimension greater than one, and real vector spaces with dimension greater than two, that there will be invariant subspaces other than {0} and V.
Maybe the book means for a particular operator?
Homework Statement
Prove or give a counterexample: If U is a subspace of V that is invariant under every operator on V, then U = {0} or U = V. Assume that V is finite dimensional.
The attempt at a solution
I really think that I should be able to produce a counterexample, however...
Homework Statement
Give a specific example of an operator T on R^4 such that,
1. dim(nullT) = dim(rangeT) and
2. dim(the intersection of nullT and rangeT) = 1
The attempt at a solution
I know that dim(R^4) = dim(nullT) + dim(rangeT) = 4, so dim(nullT) = dim(rangeT) = 2.
I also...
ok, but can I at least write a basis for null(T) and range(T)? I can't see how to prove this without defining something, because I know I can't prove this by only referring to the finite dimensions of null and range.
Homework Statement
Prove that if there exists a linear map on V whose null space and range are both finite dimensional, then V is finite dimensional.
The attempt at a solution
I *think* the following is true: For all v in V, T(v) is in range(T), otherwise T(v) = 0 which implies v is in...
Gosh, I must be getting sleepy to overlook the importance of n being unique.
So, I can show that each element of V can be written uniquely as a sum of u + n.
Should I also prove U = {au : a is in F} is a subspace of V
n and n' could definitely be different, but I don't think it matters much since they both get mapped to zero.
Is the result of a = a' is enough to prove uniqueness for a direct sum?
if V were finite dimensional then I could say, dim{null(T)} = dim(V) - dim{range(T)}.
But nothing given in the problem statement will let me assume V is finite.
Homework Statement
Suppose that T is a linear map from V to F, where F is either R or C. Prove that if u is an element of V and u is not an element of null(T), then
V = null(T) (direct sum) {au : a is in F}.
2. Relevant information
null(T) is a subspace of V
For all u in V, u is not...
Since u and v are elements of the intersection, u and v will also be elements of any subspace W that is in the intersection. And since u and v are in W and W is a subspace, this guarantees that u+v will also be in W. This same argument would apply to scalar multiplication.
Is that the...
Prove that the intersection of any collection of subspaces of V is a subspace of V.
Ok, I know I need to show that:
1. For all u and v in the intersection, it must imply that u+v is in the intersection, and
2. For all u in the intersection and c in some field, cu must be in the...
Homework Statement
Prove: If a, b are nonzero elements in a PID, then there are elements s, t in the domain such that sa + tb = g.c.d.(a,b).
Homework Equations
g.c.d.(a,b) = sa + tb if sa + tb is an element of the domain such that,
(i) (sa + tb)|a and (sa + tb)|b and
(ii) If f|a and...
Homework Statement
Let G_1 and G_2 be groups with normal subgroups H_1 and H_2, respectively. Further, we let \iota_1 : H_1 \rightarrow G_1 and \iota_2 : H_2 \rightarrow G_2 be the injection homomorphisms, and \nu_1 : G_1 \rightarrow G_1/H_1 and \nu_2 : G_2/H_2 be the quotient epimorphisms...
What are you trying to solve? You could substitute numbers for the x's and compute the value.
I don't believe there's a nifty identity for (sin(3x))^2 + (cos(3x))^2 even though it does look somewhat similar to (sin(x))^2 + (cos(x))^2.
Ironically, the book is called 'Linear Algebra Done Right' 2nd ed. by Sheldon Axler. I don't exactly love it, but it is what I'll be using this fall so I better get used to it. :rolleyes:
Thanks for all the help!
The book I'm working from does not discuss infinite dimensional vector spaces. It only gives a brief description of $\mathbf{F}$^{\infty} and P(F), the set of all polynomials with coefficents in $\mathbf{F}$.
In particular it says, "because no list spans P(F)...
Prove that $\mathbf{F}$^{\infty} is infinite dimensional.
$\mathbf{F}$^{\infty} is the vector space consisting of all sequences of elements of $\mathbf{F}$, and $\mathbf{F}$ denotes the real or complex numbers.
I was thinking of showing that no list spans $\mathbf{F}$^{\infty}, which would...
hi,
I'm doing some of the same stuff in my analysis class, and in my class notes the teacher wrote that sin(x) = O(x), not x^2.
Going back to the definitions of 'little o' and 'big O' might help.
f = O(g) means that the ratio of f/g is bounded by some constant, where f = o(g) means...
I think my confusion was with a theorem which states:
A continuous function on a compact set is uniformly continuous.
It led me to think that the real numbers (where continuous functions in previous courses have been defined) should be compact.
For some reason I was thinking I needed...
That's just the problem...
I cannot think of a sequence in \Re that does not have a limit point in \Re.
Most of my confusion is that the two statements are equivalent, but seem to imply two different things about the real numbers.
I'm assuming at this point that the real numbers are...
This is something that I think I should already know, but I am confused.
It really seems to me that the set of all real numbers, \Re should be compact.
However, this would require that \Re be closed and bounded, or equivalently,
that every sequence of points in \Re have a limit...
When I was in calculus I was taught that the only way to integrate secant cubed was by parts. Obviously this is not the case :rolleyes: .
By parts does not take too long, if you remember what to choose for u and dV!
hi,
My question reads:
Let f be defined and continuous on the interval D_1 = (0, 1),
and g be defined and continuous on the interval D_2 = (1, 2).
Define F(x) on the set D=D_1 \cup D_2 =(0, 2) \backslash \{1\} by the formula:
F(x)=f(x), x\in (0, 1)
F(x)=g(x), x\in (1, 2)...
The index of your first sum is not correct.
Remember that every time you take a derivative you loose a constant term.
After you correct your index you can then change it to something more desirable.
Look up 'hyperbolic functions' in the index of your calculus book.
In my calculus book the rules for differentiating and integrating hyperbolic functions are buried in a chapter, rather than listed on the front inside cover.
I bet your book is set up in the same way.