Subspaces Definition and 317 Threads

Homework Statement Which one of the following subsets of P_{2} (degree of 2 or below) are subspaces? a) a_{2}t^{2} + a_{1}t + a_{0}, where a_{1} = 0 and a_{0} = 0 b) a_{2}t^{2} + a_{1}t + a_{0}, where a_{1} = 2a_{0} c) a_{2}t^{2} + a_{1}t + a_{0}, where a_{2} + a_{1} + a_{0} = 2 Homework...

Homework Statement Determine which of the following sets of functions are subsets of F[a,b] a) All functions f in F[a,b] for which f(a) = 0 b) All functions f in F[a,b] for which f(a) = 1 The Attempt at a Solution Ok so I am just learning about vector subspaces. After reading the...

Eh, kind of stuck on this question. I need some suggestions on how to tackle the problem.. Homework Statement Let U and V be the subspaces of R_3 defined by: U = {x: aT * x = 0} and V = {x: bT * x = 0} (T means transpose) where a = [1; 1; 0] and b = [0; 1; -1] Demonstrate that...

Homework Statement Suppose L, M, and N are subspaces of a vector space. (a) Show that the equation L \cap (M+N) = (L \cap M)+(L \cap N) is not necessarily true. (b) Prove that L \cap (M+(L \cap N))=(L \cap M) + (L \cap N) Homework Equations N/A The Attempt at a Solution...

How do you prove that the sum of the following subspaces is R^3? U = {(x,y,z) : x - y = z} W = {(t,-t,-t) : t∈R} I guess I need to show that any vector (x,y,z)∈R^3 can be written as the sum of a vector from U and a vector from W, but I'm not sure how to do that. I know intuitively that...

Homework Statement For each of the following subsets U of the vector space V decide whether or not U is a subspace of V . Give reasons for your answers. In each case when U is a subspace, find a basis for U and state dim U Homework Equations V=P_{3} ; U=\left\{p\in\...

Hey Guys. I have some questions about vector spaces, I would really apreciate if somone could read this and let me know if I understand things or not, and if not let me know where I have it wrong. I am having a lot of trouble UNDERSTANDING how to find the intersection of two vector spaced...

Subspaces Questions Help Please! Hi I have an assignment due Monday morning and there are a few questions I am not sure about or if I proved them properly: Ok so for 2b) I said that it is not a subspace because f(x)=7 when x=0, and this function never equals zero, and since this is...

I have trouble visualizing what exactly these are. Vector Space, Subset, Sub Space... What's the difference and how can I "see" it. I'm a very visual person.

Homework Statement 1) Determine if a) (a,b,c), where b=a+c b) (a,b,0) are subspaces of R3 and 2) Determine whether the given vectors span R3 a) v1 = (3,1,4) v2 = (2,-3,5) v3 = (5,-2,9) v4 = (1,4,-1) Homework Equations - If u and v are vectors in W, then u + v is in W -...

Homework Statement Let E and F be two subspaces of R^n. Prove the following statements: (n means "intersection") If EnF = {0}, {u1, u2, ..., uk} is a linearly independent set of vectors of E and {v1, v2,...vk} is a linearly independent set of vectors Note: Above zero denotes the...

Homework Statement T = {(1,1,1),(0,0,1)} is a subset of R^{3} but not a subspace sol i have to prove it holds for addition and scalar multiplication so let x=(1,1,1) and y =(0,0,1) so x+y = (1,1,2) so it holds let \alpha = a scalar then \alphax = (\alpha,\alpha,\alpha)...

Hey guys, new to the forum here, and its midterm time and I am working through a few questions and I can't seem to figure this one out. Homework Statement Let S = { (a,b) | b > 0 } and define addition by (a,b) + (c,d) = (a*d + a*c, b*d) and define scalar multiplication by k(a,b) = (...

I have been given that V is a finite dimensional vector space over a field F and that W is a subspace of V. I need to show that v is an element of W if and only if v+w is an element of W. I know that because it is an 'if and only if' proof it needs to proved in both directions but don't...

The linear algebra course I'm taking just became very "wordy" and I am having a hard time dealing notions such as subspaces without a diagram. I was thinking Venn diagrams could be used to visualize relationships between subspaces of vector spaces. Has this been a useful way to organize the...

Homework Statement Let's say I'm given two vectors v_1 = \begin{pmatrix} 1 \\ 1 \\ 0 \\ 0 \end{pmatrix}, v_2 = \begin{pmatrix} 0 \\ 1 \\ 1 \\ 0 \end{pmatrix} \in \mathbb R^4. Let W be subspace spanned by these vectors, and define G = v_1 v_1^T + v_2 v_2^T a matrix mapping \mathbb R^4 \to...

If U_1, U_2, U_3, are subspaces of V (over fields R and/or C), is the addition of the subspaces commutative and associative? To me it seems rather trivial .. Since their summation is simply the set of all possible sums of the elements of U_1, U_2, U_3, and the elements themselves are...

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) |...

Homework Statement Prove or give a counterexample: if U1, U2, W are subspaces of V such that: U1 + W = U2 + W then U1 = U2Homework Equations The Attempt at a Solution I would be inclined to say that it's true, however I took a peek at the back of the book and that's incorrect. Here's why I...

Homework Statement Can two 4-dimensional subspaces of F62 have exactly 9 vectors in common? Can they have exactly 8 vectors in common? F62 is the 6-dimensional field where each (a1, a2, a3, a4, a5, a6) is an element of F2. The Attempt at a Solution F62 obviously has 26 = 64...

Homework Statement Hi I'm trying to prove that the sum of two subspaces U and W is also a subspace. Homework Equations U is a subspace of V if U is also a vector space and it contains the additive identity, is closed under addition, and closed under scalar multiplication. The definition of...

Homework Statement Let 1\leq r<\infty and x\in\ell_{r}=\left\{ x \text{ is a sequence with } \sum_{n=1}^{\infty}\left\vert x_{n}\right\vert^{r} \text{ converges.}\right\}, then \left\vert\left\vert x\right\vert\right\vert_{\infty}=\lim_{r\rightarrow\infty}\left\vert\left\vert...

Homework Statement H = {(x,y,z) \in R^3 | x + y^2 + z = 0} \subseteq R^3 T = {A \in M2,2 | AT = A} \subseteq M2,2 The Attempt at a Solution Our lecturer wasn't quite clear about how to go about this. He talked out closed under addition and multiplication but that's about it...

Homework Statement Okay, this is the last True/False question I will post. True or False: \text{The set of all solutions to the }m\times n\text{ homogeneous system of equations }Ax=0\text{ is a subspace of }\mathbb{R}^m. Homework Equations None The Attempt at a Solution I...

Homework Statement Prove that the intersection of any collection of subspaces of V is a subspace of V. Okay, so I had to look up on wiki what an intersection is. To my understanding, it is basically the 'place' where sets or spaces 'overlap.' I am not sure how to construct the problem...

Please anyone solve this question or can even email me on my ID abu_95bakar@yahoo.com... For the following question determine whether the set S is a sub space of the given vectorspace V. v=Rn( where n represent dimension), S is the solution set of the sysytem Ax=b, where A is an mxn...

Here we go...wheeeee Homework Statement For each of the following subsets of F3, determine whether it is a subspace of F3 (a) {(x_1, x_2, x_3) \in \mathbf{F}^3: x_1+2x_2+3x_3=0} (b) {(x_1, x_2, x_3) \in \mathbf{F}^3: x_1+2x_2+3x_3=4} (c) {(x_1, x_2, x_3) \in \mathbf{F}^3: x_1x_2x_3=0} (d)...

This is an example that I am a little confused by: U={(x,0,0)\in\mathbf{F}^3:x\in\mathbf{F}}\text{ and }W={(0,y,0)\in\mathbf{F}^3:y\in\mathbf{F}} Then U+W={(x,y,0):x,y\in\mathbf{F} Okay, I get that. Now it says that U is defined the same as above but now let...

Homework Statement (i)Show that the linear span of the vector a1 = (-7, 8, 5) is the line whose equation is x/(-7) = y/8 = z/5 The Attempt at a Solution The problem is, I don't know where or how to start.

Homework Statement Is the collection a subspace of the given vector space? If so what is the dimension? V={ax^2+bx+c: a=b+c} in P2 Homework Equations The Attempt at a Solution The first part of the question is pretty straightforward. I just verified closure under addition and...

Homework Statement Let B={b1,b2} be a basis for R2 and let T be the linear transformation R2 to R2 such that T(b1)=2b1+b2 and T(b2)=b2. Find the matrix of T relative to the basis B. The Attempt at a Solution I know that the matrix I'm looking for needs to be 2x2 and that the standard matrix...

how many 1 dimensional subspaces of Z_3^3 are there? Z_3^3 has 3^3 = 27 vectors 26 of which are non zero then we can say v and 2v have the same span and so there are in fact 13 1 dimensional subspaces. is this true?

Homework Statement "use theorem (below) to determine which of the following are subspaces of R3: (a,0,0) and (a,b,0) Homework Equations The theorem: W is a subspace of V iff: - u and v are vectors in W, u + v is in W - k is a scalar, u is a vector in W, then ku is in W...

Homework Statement Which of the following subsets of R3? The set of all vectors of the form a) (a, b, c), where a=c=0 b) (a, b, c), where a=-c c) (a, b, c), where b=2a+1Homework Equations A real vector space is a set of elements V together with two operations + and * satisfying the following...

Is there a linear space V in which the union of any subspaces of V is a subspace except the trivial subspaces V and {0}? pls help

Alright. As stated in the title above. So, a subspace is a set of vectors that satisfies: 1) It contained the zero vector; 2) It's closed under addition and subtraction. By "closed", it means that when I add another vector in R2 or multiply by a scalar k on A(x)=m, it will end up with...

Homework Statement Let V be a finite dimensional, nonzero complex vector space. Let T be be a linear map on V. Show that V contains invariant subspaces of dimension j for j=1, ..., dim V. Homework Equations Since V is complex, V contains an invariant subspace of dimension 1. The...

Vector Subspaces, don't understand... Homework Statement Which of the given subsets of the vector space, M23, of all 2 X 3 matrices are subspaces. (a) [a b c, d 0 0] where b = a + c Homework Equations Theorem 4.3 Let V be a vector space with operations + and * and let W be a...

Homework Statement True/false: Union of two vector subspaces is a subspace. Homework Equations none The Attempt at a Solution I'm unsure if this is true because I'm also unsure if it already assumes that it is closed under scalar multiplication and addition. If it is closed, then...

If P_{5}(x) is the set of all polynomials in x in degree less than 5. Which of following subsets of P_{5}(x) are subspaces. (i) the set of all polynomials in P_{5}(x) of even degree (ii) the set of all polynomials in P_{5}(x) of degree 3 (iii) the set of all polynomials p(x) in P_{5}(x) such...

Homework Statement I need to find the basis of W1 + W2 and W1 intersect W2 (It's part of a larger homework problem that I know how to do, but I am stuck on the trivial step...per usual) W_1 = \left(\begin{array}{c c} x & -x \\ y & z \end{array}\right) W_2 = \left(\begin{array}{c c} a...

i am given 2 subspaces of R4 W=sp{(a-b,a+2b,a,b)|a,b\inR} U=sp{(1,0,1,1)(-6,8,-3,-2)} and am asked to find: a homogenic system for W- system for a vector (x,y,z,t) belonging to W i see the basis for W is : a(1,1,1,0)+b(-1,2,0,1),, i put these vectors into an extended matrix with (x y z...

Statement Let S be a linear operator S: U-> U on a finite dimensional vector space U. Prove that Ker(S) = Ker(S^2) if and only if Im(S) = Im(S^2) So, I'm really not sure about how to prove this properly. I have a few ideas, but this one seemed to make sens intuitively to me. So, I'm...

Homework Statement Let H and K be subspaces of a vector space V. Prove that the intersection K\cap H is a subspace of V. Homework Equations The Attempt at a Solution This, I have absolutely NO idea how and where to start. Are there any axioms which can be used to prove this?

What exactly is the "addition of subspaces?" It is obviously not the same as the "union of subspaces," since the union of subspaces A and B in V is a subspace of V only if A is contained in B (or B is contained in A).

1. Suppose that U is a subspace of V. What is U+U? 2. Homework Equations : There's a theorem that states: Suppose that A and B are subspaces of V. Then V is the direct sum of A and B (written as A [plus with a circle around it] B) if and only if: 1) V=A+B (meaning, the two...

Homework Statement Let V be a vector space and U a subspace of V . For a given x ∈ V , define T= {x + u | u ∈ U }. Show that T is a subspace of V if and only if x ∈ U . Homework Equations Subspace Test: 1: The 0 vector of V is included in T. 2: T is closed under vector addition 3...

I have a question about this theorem. Let V be an n-dimensional inner product space, and let T:V-->V be an orthogonal linear transformation. Let S be a minimal invariant subspace under T. Then S is one dimensional or two dimensional. I understand what this theorem says and I follow the...

i am given these 2 groups W=sp{(1 0 2 0) (1 1 1 1) (1 0 0 0)} U=sp{(1 0 1 1) (1 2 1 2) (0 0 1 0)} and am asked to find a basis for each one and their dimention a basis for W+U a basis for W\capU ----------------------------------------------- for the basis i found that they are...

Homework Statement Show that the following are subspaces of R^m : (a) The set of all linear combinations of the vectors (1,0,1,0) and (0,1,0,1) (of R^4) (b) The set of all vectors of the form (a,b,a-b,a+b) of R^4 Homework Equations The Attempt at a Solution (a) If...