Vector spaces Definition and 286 Threads

Here is a question I have been given: V3(R) represents the set of vectors in 3-dimensional space. What kind of geometrical objects are represented by the various subspaces of V3(R)? For instance the 1-dimensional subspace S with basis { (0, 1, 0)T } represents the set of vectors parallel to...

Find the dimension of the subspace spanned by the vectors u, v, w in each of the following cases: i) u = (1,-1,2)^T v = (0,-1,1)^T w = (3,-2, 5)^T ii) u = (0,1,1)^T v = (1,0,1)^T w = (1,1,0)^T Right, how do I go about this, do I have to find the subspace first then do the dimension. Can...

Show that the isomorphism O:L2(E)—>(E,E*) Where E is a vectorspace over a field K E* is a dual space L2:bilinear form L: n-linear form.

Say I have to vector spaces V,W and a linear transformation \Phi:V\rightarrow W. I know that (given v,p\in V) if I interpret a tangent vector v_p as the initial velocity of the curve \alpha(t)=p+tv I have, relative to a linear coordinates system on V, v_p=x^i(v)\partial_{i(p)}. The thing I don't...

Whilst trying to refresh myself on what a dual space of a vector space is I have confused myself slightly regarding conventions. (I am only bothered about finite dimensional vector spaces.) I know what a vector space, a dual space and a basis of a vector space are but dual bases: I seem...

NEVER MIND, FIGURED THEM OUT Definitions (All vector spaces are over the complex field) If \mathcal{M} is a subspace of a normed vector space (\mathcal{X}, ||.||_{\mathcal{X}}) then ||x + \mathcal{M}||_{\mathcal{X}/\mathcal{M}} =_{def} \mbox{inf} _{m \in \mathcal{M}}||x + m|| defines...

Hi, I recently bought a new linear algebra book. I've been through the subject before, but the book I had back then glossed over the abstract details of vector spaces, amongst other things. However, I've found something questionable in the first few pages, and I was wondering if someone could...

I'm tutoring a linear algebra/diff eqs class and we are about to start on vector spaces; the point is this, I would like to present them with a variety of unsual vector spaces (along with the usual ones) that they may understand that vectors are not just directed line segments, but rather more...

Er that's it really. In the various texts I've got that introduce vector spaces, they always say 'defined over a field' and then give R and C as examples. Are there other fields that mathmos or physicists define vector spaces over? Does a vector space satisfy the axioms of a field? I'll think...

1. let V be a vector space, U1,U2,W subspaces. prove/disprove: if V=U1#U2 (where # is a direct sum) then: W=(W^U1)#(W^U2) (^ is intersection). 2. let V be a vector space with dimV=n and U,W be subspaces. prove that if U doesn't equal W and dimU=dimW=n-1 then U+W=V. for question two, in...

hi, I am confused about vector spaces and subspaces. I've just started a book on linear algebra, and i understood the 1st chapter which delt with gaussian reduction of systems of linear equations, and expressing the solution set as matricies, but the 2nd chapter deals with vectors and I'm...

Vector Geometry and Vector Spaces... Hi - I've just started my degree course at university, studying theoretical physics. However, I have opted to attend the same maths lectures that some of the mathematics students are taking. We have been learning about "geometry and vectors in the plane"...

i've been having some trouble with my linear algebra homework and I am wondering if you guys could give me some insight or tips on these problems: Let v be any vector from V, and let a be any real number such that av=0. Show that either a=0 or v=0. - i was thinking about assuming the...

i'm really really confused abt vector spaces and how to prove if something is a vector space :confused: Could som1 please help! Example: why is V = {(1 2)} not a vector space??

{1, x, 2x^2} is a basis for V (the polynomial vector space with maximum power 2) then could I say that the coordinate vectors with respect to V, which form the set {(1,0,0), (0,1,0), (0,0,2)} for R^3 is equivalent to the above set in V? Although the word equivalent is not defined. But it...

Lately I've been taking a unit that deals with abstract algebra and I'm finding myself not understanding the lectures at all. To make matters worse the unit doesn't have a reccomended textbook so I don't even have any infomation to self learn from. I guess what I'm asking is for some good...

"Let T be a linear transformation on a finite dimensional real vector space V. Show that T is diagonalisable if and only if there exists an inner product on V relative to which T is self-adjoint." The backward direction is easy. As for the forward direction, I don't understand how given an...

(Urgend)Basis and Vector Spaces (Need review of my proof) Hi I'm trying to proof the following statement: Here is my own idear for a proof that the set of vectors v = \{v_{1}, v_{1} + v_{2},v_{1} + v_{2} + v_{3} \} Definition: Basis for Vector Space Let V be a Vector Space. A set...

Determine if this is a vector space with the indicated operations the set of V of all polynominals of degree >=3, togehter iwth 0, operations of P (P the set of polynomials) now all the scalar multiplication axioms hold. the text however says that the axion \mbox{For u,v} \in V, \mbox{then} \...

Do you ever think up theorums and think: "that's inetersting, I wonder if anyone's ever thought of that before?" In this vein the other day, I thougt up these two. They are both fairly trivial and possibly it's only me that finds them worth even bothering with, but what I want to know is if...

Is it possible for a nullspace to be spanned by only one vector? Does a statement like Nul A = Span{\vec{v}_1} even have any meaning?

(Or if you prefer: Why are things defined this way?) I noticed that, in my book's definition, scalar multiplication (SM) on vector spaces lacks two familiar things: commutativity and inverses. The multiplicative inverse concept doesn't seem to apply to SM. Can it? I can't imagine how it could...

Vector Spaces, Subspaces, Bases etc... :( Hello. I was doing some homework questions out of the textbook and i came across a question which is difficult to understand, could somebody please help me out with it? -- if U and W are subspaces of V, define their intersection U ∩ W as follows...

Hi, I'm having trouble with these homework questions. I have to prove that B*0v = 0v , where B is a scalar. Also, I have to prove that if aX = 0v , then either a = 0 or X = 0 ---where a is a scalar and X is a vector. I know that I have to use the 8 axioms but I'm not sure...

I have a homework problem that I can't figure out and there is nothing in the book that helps me out. I was hoping someone could shed some light. Let R^+ denote the set of postive real numbers. Define the operation of scalar muplication, denoted * (dot) by, a*x = x^a for each X...

what is the reason behind choosing the linear vector spaces in representing the state of a system? why is it convenient ? and why do we actually need a linearity ?

Suppose I have a set involving trigonometric functions, with addition defined as multiplication of two vectors. If this is a vector space, the zero vector has to be unique. If cos (0) works as the zero vector, then cos (2*pi), etc. also work. Does this mean the set is not a vector space...

hi,i got 2 question about vector spaces : 1. Do the set of all n-tuples of real numbers of the form (x, x1 ,x2...xn) with the standard operation on R^2 are vector spaces? 2.Do the set of all positive real numbers with operations x+y =x*y and kx=x^2 are vector space?

I have some questions concerning the rationnal numbers and the vector spaces. Let's take the set of rational number Q with the usual addition and multiplication. We can say that (Q,+,.) is a vector space on the Q field. Now, if we add the |x| absolute value, we define have vector space with...

hi this problem requires calculus, as it also concerns Vector spaces, i solved a lot of Vector spaces problems, either the subset is matrix or ordered pairs. this question says: Which of the following subsets of the vector space C(-inf, inf) are subspaces: (note: C(-infinity, infinity)...

I am in need of some guidance on a question concerning vector spaces and spanning sets. Q) Suppose that V is a vector space over F and {v1,...,vn} ⊂ V. a) Prove that if {v1,...,vn} spans V, then so does {v1-v2,v2-v3,...,v(n-1)-v(n),vn}.

I'm really confused about a question I came across in my textbook. It basically says this: Consider the set of polynomial functions of degree 2. Prove that this set is not closed under addition or scalar multiplication (and therefore not a vectorspace). I'm confused because I think it is...

I hope someone can help me (guide) in this theorem. How can I show that a "subset W of a vector space V is indeed a subspace of V if and only if given u and v as vectors in W and a and b are said to be scalars, then au + bv is in W."? Can I assume a vector with my desired number of...

Let V denote the set of all differentiable real-valued functions defined on the real line. Prove that V is a vector space with the operations of addition and scalar multiplication as follows: (f + g)(s) = f(s) + g(s) and (cf)(s) = c[f(s)] --------------- I know I have to prove this by...

Is the same thing a linear space and a vector space?

I have an upcoming Linear Algebra exam and my textbooks are really vague in defining certain concepts (and he didn't limit the ambiguous nature to Linear Algebra. His Calculus book is the same way). Would someone mind helping me define or determine Determination tests for concepts like...