Subspace Definition and 560 Threads

Homework Statement Given W={A belonging to M2(ℂ) | A is symmetric} is a subspace of M2(ℂ) over ℂ, when showing it is closed under scalar multiplication, do I need to use a complex scalar as it is over the complex numbers, or will a real number be okay? Homework Equations The...

Let ##T: V → V ## be a linear map on a finite-dimensional vector space ##V##. Let ##W## be a T-invariant subspace of ##V##. Let ##γ## be a basis for ##W##. Then we can extend ##γ## to ##γ \cup S##, a basis for ##V##, where ##γ \cap S = ∅ ##, so that ## W \bigoplus span(S) = V ##. My question...

I've been thinking about a problem I made up. The solution may be trivial or very difficult as I have not given too much thought to it, but I can't think of an answer of the top of my head. Let ## T:V → V ## be a linear operator on a finite-dimensional vector space ##V##. Does there exist a...

(All that follows assumes we are talking about a self-adjoint operator A on a Hilbert space \mathscr H.) The first volume of Reed-Simon defines \mathscr H_{\rm pp} = \left\{ \psi \in \mathscr H: \mu_\psi \text{ is pure point} \right\}. The book seems to take for granted that \mathscr H_{\rm...

I'm self-studying Linear Algebra and the book I'm using is Linear Algebra done right by Sheldon Axler but I came across something that I don't understand .- Suppose \mathrm U is the set of all elements of \mathbb F ^3 whose second and third coordinates equal 0, and \mathrm W is the set of all...

Homework Statement The original printed problems can be found as attachments. The questions ask if a set S is a subset Rn. Give Reasons Question 1.) S is the set of all vectors [x1,x2] such that x12 + x22 < 36Question 2.) S is the set of all vectors [x1,x2,x3] such that: x2= 2x1 x3 = 3x1...

Homework Statement Let W be a subset of vector space V. Is it s subspace as well? W = {(a1, a2, a3) \in ℝ3 : 2a1-7a2+a3=0}So, to check if this is a subspace I need to satisfy the following: 1. That 0 is in the set. Plugging (0,0,0) into the equation 2a1-7a2+a3=0 yields 0=0 so yes, it is...

Homework Statement Show that a subset W of vector space V is a subspace of V iff span(W) = V The Attempt at a Solution OK, I am trying to see if my reasoning is correct or if I am overthinking this. To show this is a subspace three things have to be true. (a) 0 \in W, (b) vectors x + y...

Hello, I want ask some subspace problems. Attachment is a question. contains all polynomials with degree less than 3 and with real coefficients. I want prove that item 1 and item 2 are subspace or not. Am I insert real number to the item 1& 2 equation to test as follows: (a) 0 ∈ S. (b) S is...

Hi guys, back again...Any help on this question would be appreciated. Thanks in advance :)

Here is the question: Here is a link to the question: Orthogonal basis? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement Determine whether the set of all vectors of the form (sin2t,sintcost,3sin2t) is a subspace of R^3 and if so, find a basis for it. Homework Equations I guess you just need to use the axioms where it is closed under scalar addition and multiplication. The Attempt...

Let L^2 be the usual vector space of complex sequences. Let F be the subspace of sequences whose first term is zero. Show that F is closed. Let $((V_{nk}):k=1,2,...)$ be a convergent sequence in F. I need to show it converges to a sequence whose first term is 0. Well, for all positive...

For vector such as <a,1,1>, to prove its a subspace in r3, is it alright to immediately assume its not a subspace, as it doesn't meet the zero vector condition in that, 1=\=0? And is there a specific way to set out subspace questions? I seem to just use intuition and two or three lines which...

Homework Statement 1. Let A= {a,b,c}. Calculate the subspace topology on A induced by the topology T= { empty set, X,{a},{c,d},{b,c,e},{a,c,d},{a,b,c,e},{b,c,d,e},{c}, {a,c}} on X={a,b,c,d,e}.Homework Equations Given a topological space (X, T) and a subset S of X...

Homework Statement Determine whether W is a subspace of the vector space: W={(x,y):y=ax, a is an integer} , V=R^2 Homework Equations noneThe Attempt at a Solution Is u+v in W? Let u = (u,au) and v = (v,av) u+v = (u,au) + (v,av) = (u+v, au + av) = (u+v, a(u+v)) If x = u+v => u + v = (x,ax) =>...

Homework Statement Hi, This is my first post. I had a question regarding open/closed sets and subspace topology. Let A be a subset of a topological space X and give A the subspace topology. Prove that if a set C is closed then C= A intersect K for some closed subset K of X. Homework...

Homework Statement 1) The set H of all polynomials p(x) = a+x^3, with a in R, is a subspace of the vector space P sub6 of all polynomials of degree at most 6. True or False? 2) The set H of all polynomials p(x) = a+bx^3, with a,b in R, is a subspace of the vector space P sub6 of all...

I can't seem to work out this question because it's so weird The set F of all function from R to R is a vector space given the diffential equation f"(x)+3f'(x)+x^2 f(x) = sin(x) is a subspace of F? Justify your answer I know that we have to proof that it's non-empty 0. The zero vector has to...

Homework Statement There is a vector space with set F, of all real functions. It has the usual operations of addition of functions and multiplication by scalars. You have to determine whether this equation is a subspace of F: f''(x) + 3f'(x) + x^2 f(x) = sin(x) Homework Equations f''(x) +...

Let U be a subset of ℝn be an open subset and let f:U→ℝk be a continuous function. the graph of f is the subset ℝn × ℝk defined by G(f) = {(x,y) in ℝn × ℝk : x in U and y=f(x)} with the subspace topology so I'm really just trying to understand that last part of this definition...

Homework Statement Find bases for the following subspace a of r^3 Y+z=0 The Attempt at a Solution First I found a normal to this plane n=(0,1,1) Then I found two vectors which are orthogonal to the normal u=(0,-1,1), v=(1,0,0) Is this correct the answer in my book has...

Homework Statement Determine whether the following is a Subspace of r^3: All vectors of the form (a,b,c), where b=a+cThe Attempt at a Solution The answer in the book says it is not a subspace but I can only find examples that show it is a Subspace I.e. Let u=(a,a+c,c)=(1,2,1)...

Homework Statement Is the following set a subspace of R2? Homework Equations (a,b2) The Attempt at a Solution I'm exhausted and stumped.

Homework Statement Let H= C[-1,1] with L^2 norm and consider G={f belongs to H| f(1) = 0}. Show that G is a closed subspace of H. Homework Equations L^2 inner product: <f,g>\to \int_{-1}^{1}f(t)\overline{g(t)} dt The Attempt at a Solution I've been trying to prove this for a...

Homework Statement Prove that if S is a subspace of R1, then either S={0} or S=R1. Trying to come up with a proof I dissected each statement, I know that in order for S to be a subspace the zero vector must lie within the subset. So I know S={0} is true. I then checked an arbitary...

Homework Statement {(x1,x2)T| |x2|=|x2|} So my first thought is we would have to check for both cases (x1,x1) and (-x1,-x1) a=(x1,x1)T b=(v1,v1)T βa=(βx1,βx1)T for the case where a<0 βa(-βx1,-βx1)T thus it is closed under scalar multiplication. a+b=(x1+v1,x1+v1) for the case...

Homework Statement Determine whether the following sets form subspaces of R2 {(x1,x2)T|x1=3x2} So I rewrote the set in order for it to be homogenous ( I'm not sure why we would do that but I saw a problem saying if we can do it to do it. {(x1,x2)T|x1-3x2=0} So my logic when...

Homework Statement I am considering the space \tilde{W}^{1,2}(\Omega) to be the class of functions in W^{1,2}(\Omega) satisfying the property that its average value on \Omega is 0. I would like to show that \tilde{W}^{1,2}(\Omega) is a closed subspace of W^{1,2}(\Omega). Homework...

Here is the question: Here is a link to the question: Need help proving R(T), the image subspace is T-invariant.? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement Determine whether the following sets form subspaces of R^{2} A){(x_{1},x_{2})^{T} | x_{1}x_{2}=0} B){(x_{1},x_{2})^{T} | x_{1}=3x_{2}} Homework Equations checks: Does zero vector exist? Is the space closed under addition? Is the space closed under scalar multiplication?The...

Homework Statement Is U = {A| A \in nℝn, A is invertible} a subspace of nℝn, the space of all nxn matrices? The Attempt at a Solution This is easy to prove if you assume the regular operations of vector addition and scalar multiplication. Then the Identity matrix is in the set but 0*I and...

Find a basis for the subspace S = span{(1,2,1,2,1) , (1,1,2,2,1), (0,1,2,0,2)} of Z53 (The set of elements in the field of modulus 3) Attemept: So the issue isn't in finding a basis per say. If this was the field of Real numbers I wouldn't have an issue, I would just row reduce and use the...

This Exercise 3.3 from Advanced Calculus of Several Variables by C.H. Edwards Jr.: [SIZE="3"]If V is a subspace of \Re^{n}, prove that V^{\bot} is also a subspace. As usual, this is not homework. I am just a struggling hobbyist trying to better myself on my own time. The only progress I've...

Homework Statement Let L,M,N be subspaces of a vector space V Prove that (L \cap M) + (L \cap N) \subseteq L \cap (M + N) Give an example of subspaces L,M,N of \mathbb{R}^2 where (L \cap M) + (L \cap N) \neq L \cap (M + N) Homework Equations The Attempt at a Solution...

Homework Statement When is it true that the only subspaces of a vector space V, are V and {0}? Homework Equations NA The Attempt at a Solution Because a subspace has to be closed under addition and scalar multiplication, it is my intuition that this is true only when there are no infinite...

Hi, I'm trying to understand the fiber bundle formulation of gauge theory at the moment, and I'm stuck on the connection. Every reference I've found introduces the idea of a connection on a principle bundle as a kind of partitioning of the tangent space at all points in the total space into a...

I have a (3 x N) matrix of column rank 2. If each column is treated as a point in 3-space, then connecting the points draws out some planar shape. What operation can I apply such that this planar shape is transformed onto the x-y axis, so that the shape is exactly the same, but is now described...

Homework Statement Task: Show that U = {(x, y) | xy ≥ 0} is not a subspace of vector space R2 I wish you could help me to understand why U is not a subspace of R2x2. I have actually found a vectors u and v such that it does not belong to U (e.g. (-3,-1) +(2,2) = (-1,1) ) but is that...

Homework Statement Let A be an n*n matrix and let B be a real number, Nothing which properties of matrix multiplication you use, show that the set W= {x member of R^n | Ax=Bx} is a subspace of R^n, where x in the equation Ax=bx is represented by a column vector instead of an n-tuple with...

Hi We have a linear transformation g : ℝ^2x2 → ℝ g has U as kernel, U: the 2x2 symmetric matrices (ab) (bc) A basis for U is (10)(01)(00) (01)(10)(01)I thought this would be easy but I've been sitting with the problem for a while and I have no clue on how to solve it...

Ok, so I understand that a vector space is basically the span of a set of vectors (i.e.) all the possible linear combination vectors of the set of vectors... I don't understand the concept behind a subspace or why it's useful. I know the conditions are: 1. 0 vector must exist in the set...

I am trying to visualize the subsppace topology that is generated when you take the Rationals as a subset of the Reals. So if we have ℝ with the standard topology, open sets in a subspace topology induced by Q would be the intersection of every open set O in ℝ with Q. Since each open set...

I attached the problem, the solutions say its not a subspace. To be a subspace it must satisfy 3 conditions 1) 0 is in S 2) if U and V are in S, then U+V must be in S 3) if V is in S, then fV is in S for some scalar f. 0 is in S U+V is in S because if U and V have elements that are...

Which of the following is a subspace of M2x2 (the vector space of 2x2 matrices. and explain why or why not: 1) Set of 2x2 matrices A such that det(A)=1 2) set of 2x2 matrices B such that B[1 -1]^t=0 vector To check if something is a subspace I must satisfy 3 conditions (applied for matrix A): 1)...

The problem is attached. I'm having problems with parts a and c, well maybe not part a (probably just need to check if I did this part right. I'm just not sure if I'm wording part a right. Anyways for part a I must prove it's a subspace so I must satisfy 3 conditions: 1) 0 is in S 2) if U and...

Hi. I'm trying to check if my approach is right. The problem is attached. I need to check these: 1) 0 vector is in S 2) if U and V are in S then U+V is in S 3) if V is in S, then cV where c is a scalar is in S The 1st condition is not satisfied right? Since A*[0 0]^t=[0 0]^t≠[1 2]^t?

The problem is attached. I need to determine if it's a subspace. So it must satisfy 3 conditions: 1) 0 vector is in S 2) if U and V are in S then U+V is in S 3) if V is in S, then cV where c is a scalar is in S For the 1st condition: I said the 0 vector is not in S, but I'm not sure if...

Homework Statement Let S be the linear span of the orthogonal set: {[3 2 2 2 2]T,[2 3 -2 -2 -2]T,[2 -2 3 -2 -2]T} Calculate the orthogonal projection of Y = [1 2 -1 3 1]T onto S. The Attempt at a Solution Not sure how to go about this... Do i find a vector that is orthogonal...

Homework Statement V is a vector space The Attempt at a Solution If S is smaller than V then there exists a T such that S + T = V. OTHERWISE S = V. I'm not sure what assumptions am I making which I could break down to prove...