# Tagged Questions

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### Problems on vector spaces

Let $E$ a $\mathbb{K}$-vector space of finite dimension $n$, $\mathcal{V}$ a subspace of $\mathcal{L}(E)$ such that $$\forall u\in\mathcal{V}\setminus \{0\},u\in\mathcal{GL}(E)$$ a) Show that ...
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### Solution to homogeneous linear differential equation form a vector space

Show that the solutions of a homogeneous linear differential equation $y"+a(x)y'+b(x)y = 0$ form a vector space. What is its dimension? I understand that the dimension is 2 and that 0 is a solution ...
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### Computing intersection of vector spaces spanned by two lists

Assume that I'm given two lists of vectors $l_1$ and $l_2$, where all the vectors have equal dimension. I want to compute a basis for the intersection of their spans. What is the easiest setup for ...
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### Subsets that are also vector spaces

The vector space $R^3$ and the subset M consists of the vectors $(\xi_1,\xi_2,\xi_3)$ for which i) $\xi_1 = 0$ ii) $\xi_1 = 0$ or $\xi_2 = 0$ iii) $\xi_1 + \xi_2 = 0$ iv) $\xi_1 + \xi_2 = 1$ ...
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### Function from one Null space to Another

Suppose a single vector space over $R$ of degree $n$, and two matrices $A, B$ of arbitrary row size, but col size $n$, s.t. their individual null spaces are linear subspaces of this vector space. Is ...
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### For a linear function, the fiber of the output is the translate of the kernel by the input. (Trivial observation, proof needed.)

As you may already know, I am a newbie to linear algebra. I am supposed to prove that for every linear function between vector spaces, for every input, the fiber of the corresponding output equals the ...
Let $A(2,-1,1)$, $B$ and $C$ be the vertices of a triangle where $\overrightarrow{AB}$ is parallel to \vec{v}=(2,0,-1), $$\overrightarrow{BC} is parallel to \vec{w}=(1,-1,1) and \angle(BAC)=90°. ... 1answer 36 views ### Question about dimension of a subspace Let K be a field and define the following subspaces$$V=\textrm{span}(e_1,e_2,e_3),\;\; V^\bot = \textrm{span}(e_4,e_5,e_6)$$inside K^6. Let \dim L=4 and assume that \dim L\cap V\leq 1. Can ... 0answers 43 views ### Basis of \mathbf{Q}[x] I wanna show that the binomials \binom{x}{k} for k=0,1,\ldots form a basis of the \mathbf{Q}-vector space V=\mathbf{Q}[x]. I can show that for fixed m\in\mathbf{N} the \binom{x}{k} ... 0answers 13 views ### Use of Matlab to put equation into vector form Is there a way to put the following equation of a line into vector form using Matlab? \displaystyle y=\frac{cos(s_n)-cos(s_{n+1})}{sin(s_{n+1}-sin(s_n)}(x-sin(s_n))-cos(s_n) 1answer 24 views ### Deduction of vector form of Snell's law I was unable to find the deduction of the vector form of Snells's law.$$n_1\sin\theta_1 = n_2\sin\theta_2$$Here is the vector form, from the article A Theory of Multi-Layer Flat Refractve Geometry ... 0answers 26 views ### linear algebra question Consider n convex polytopes S_1, \cdots, S_n and a set of matrices W such that each matrix A\in W, we have that the i-th row of A is a member of S_i. (In general W is infinite.) ... 1answer 34 views ### dim_\mathbb C V=n then dim _\mathbb R V=2n Prove that if the dimension of a vector space V over \mathbb C is n then the dimension of V over R is 2n I wanted to do it using isomorphisms i.e. every finite dimensional vector space ... 0answers 18 views ### How to prove for an operator L on a vector space V that Null(L^k)\subset Null(L^{k+1})? This was a past exercise and I still struggle to understand why it is necessary to prove it. I could very well be doing it wrong too! We have L, an operator (I'm assuming linear, but feel free to ... 1answer 17 views ### A Quotient space Problem Prove that there is a natural isomorphism between (V/W)' and W^0 where W^0 is the annihilator of W and (V/W)' is the dual of V/W 2answers 47 views ### Prove that L(V,W) forms a vector space Let V and W be vector spaces over a field F. Let L(V,W) = \{T:V\to W : T \text{ is linear} \}, that is, L(V,W) is the collection of all linear functions from V to W. For S,T \in L(V,W) ... 1answer 35 views ### Hamel Basis Exercise Proof Clarification. While looking up something else on stack exchange, I ran across this question An exercise about a Hamel basis and it intrigued me. The answer was provided by Jonathan Golan ... 0answers 38 views ### Linear Algebra (Basis) We have B: Question: Find a basis in \mathbb{M}_{3,2}(\mathbb{R}) that has B. Obs.: I have no idea how to do this. I know that a combination of a basis is a vector in the subspace formed for this ... 0answers 17 views ### Linear algebra (Coordinates) Question: Find the coordinates of x=(1,0,0) in relation to base$$B=\{(1,1,1),(-1,1,0),(1,0,-1)\}.$$I tried: a,b,c\in R such that$$a(1,1,1)+b(-1,1,0)+c(1,0,-1)=(1,0,0)=x$$but I'm not sure ... 1answer 21 views ### Is this an Alternative Proof of a set of vectors forming a basis? This is one of my exam past paper question So I proved this correctly by following the normal method which is showing that a, b and c are linearly independant My proof - When I looked at the ... 1answer 30 views ### Find a basis of M_2(F) so that every member of the basis is idempotent Let V=M_{2\times 2}(F) (the space of 2x2 matrices with coefficients in a field F). Find a basis \{A_1,A_2,A_3,A_4\} of V so that A_j^2=A_j for all j. My attempt. Let A_j be ... 1answer 36 views ### Nonhomogeneous Linear Systems and Vector Space Solutions Are there any nonhomogeneous linear systems with a solution set that forms a vector space? I know that, in order to be a vector space, a set must consists of a set V together with operations + (called ... 1answer 33 views ### being \mathbf{w} a vector, how do I calculate the derivative of \mathbf{w}^T\mathbf{w}? Let's say that I have a vector \mathbf{w}. How can I calculate the derivative in the following expression? \frac{\mathrm{d}}{\mathrm{d}\mathbf{w}}\mathbf{w}^T\mathbf{w} 0answers 25 views ### being \mathbf{a} and \mathbf{b} two vectors with same length, how do I expand (\mathbf{a}^T\mathbf{b})^2? Let's say that I have two vectors \mathbf{a} and \mathbf{b}. Assuming that they have same length, their product \mathbf{a}^T\mathbf{b} and its square (\mathbf{a}^T\mathbf{b})^2 are scalars. ... 1answer 24 views ### Proof linear independency lemma If \mathbf{u} and \mathbf{v} is in the complex vector space V and \mathbf{w}_1 = \mathbf{u} + i \mathbf{v} and \mathbf{w}_2 = \mathbf{u} - i \mathbf{v} are linear independent then will the ... 1answer 29 views ### Transformation from cartesian to polar Coordinates of Vector Field This is fairly low-level, still I would like to get a verification: I vector field$$\mathbf{F}=F_x \hat{e_x} + F_y \hat{e_y} = F_r \hat{e_r} + F_{\phi} \hat{e_\phi}$$given in cartesian coordinates, ... 0answers 29 views ### Relation between basis elements under automorphisms Let e_1, e_2, e_3 denote the standard basis for the vector space \mathbb{R}^3, and let f, g: \mathbb{R}^3\to \mathbb{R}^3 be linear maps such that g\circ f = f\circ g = {\rm id}. Also, let ... 0answers 105 views +50 ### Vector space basis change: is this “index-free” notation correct? There are already quite a number of questions on basis change in a vector space. Nevertheless, to fully grasp the underlying idea I made up the following notation and I have some doubts on it (note: ... 0answers 19 views ### A question on existence of a linear map on finite dimensional vector spaces Suppose that V, W are finite dimensional vector spaces and U a subspace of V such that \dim U\ge \dim V-\dim W , then how do we prove that there is a linear map T:V\to W such that \ker ... 1answer 40 views ### Finding set of vectors that spans the solution set Question: Find a set of vectors \{u,v\} in \mathbb{R}^4 that spans the solution set of the equations:$$\begin{align}w - x + y + z = 0 \\ 5w + 2x - y + z = 0\end{align}$$Reducing these I get: ... 0answers 36 views ### normal vectors in spaces where n > 3 I am reading Lovelock and Rund's book on Tensors and they make a statement that I wanted to validate about normal vectors in high-dimensional spaces. It should be remarked that the above ... 0answers 30 views ### Proving basis of \mathcal{L}(V,W) Suppose two vector spaces V and W over some field F is given. Now let \mathcal{L}(V,W) be the set of all linear maps from V to W. Also let \dim V=n,\dim W=m and \mathscr{M}(m,n,F) be ... 2answers 56 views ### General vector space theory developed without matrix-theory. Since vector spaces can exist regardless of a matrix I wanted to see if we could do all the proofs for the general vector-spaces without using theory for matrices. Then it was only two proofs of the ... 1answer 47 views ### Does (x,f(x),\cdots,f^p(x)) is linearly dependent over E implies (id, f, …, f ^ p) is linearly dependent over \mathcal{L}(E)? Here is the original (classic I think) problem I had encored: if (x,f(x)) is a linearly dependent family of E (a vector space) for all x\in E, then the family (id,f) is linearly dependentt ... 0answers 30 views ### Prove that if m<n then S does not generate V Let V be a vector space over a field K such that dim V=n and let S\subseteq V such that |S|=m. Prove that if m<n then S does not generate V Let S={s_1,...,s_m}. Suppose that ... 1answer 23 views ### definition of linearly dependent set I know that this is a silly question to ask but I would really appreciate if you can answer me. Let V be a vector space and S=\{v_1,\ldots,v_n\} a finite subset of V. S is linearly dependent ... 2answers 53 views ### A question on dual spaces of vector spaces I've been doing a bit of self study into the formalism of dual spaces. In the book that I've been reading the author introduces the notion of a dual space, V^{\ast} to a given vector space V as ... 1answer 25 views ### Linear algebra perpendicular vectors How do I know the vectors that are perpendicular to (1,1,1) and (1,2,3) lie on a line?, thanks beforehand, I'll appreciate any help here. 1answer 20 views ### Sub-space problem in complex space A problem in my professor's guide is driving me nuts, I don't even know where to start, this is the problem: Is \{(z,u) \in \mathbb{C}^2 / z - \overline{z} + u = 0\} a sub-space of ... 2answers 30 views ### Rank of I_m - X_{m \times m} given rank of X I have a matrix X_{m\times m} which is idempotent and has rank(X) = n < m. I have for some time now been trying to calculate rank(I_m - X) but have been unable to do so. I should be able to ... 1answer 29 views ### Linear operators and change of basis in a vector space Suppose we have a vector space V over a scalar field \mathbb{F} and two different bases \mathcal{B}=\lbrace\mathbf{v}_{i}\rbrace_{i=1,\ldots , n} and ... 4answers 24 views ### Let X,Y,Z be subspaces of V so that X is a subspace of Y. Prove that Y\cap (X+Z)=X+(Y\cap Z) Let X,Y,Z be subspaces of V so that X is a subspace of Y. Prove that Y\cap (X+Z)=X+(Y\cap Z) I know that I need to prove that Y\cap (X+Z)\subseteq X+(Y\cap Z) and X+(Y\cap Z)\subseteq ... 2answers 34 views ### What does it mean when dim(V)=rankT I have a question relating to a linear transformation and have ended up with the result that dim(V)=rank(T). I got to this because I'm told that V and W are finite dimensional vector spaces, ... 3answers 27 views ### Prove that \exists y\in V so that the set {{u+y:u\in U}} is a subspace of V Let V a vector space over a field F, and let v,w\in V so that v\neq w. Define U={{(1-t)v+tw: t\in F}}. Prove that \exists y\in V so that the set {{u+y:u\in U}} is a subspace of V ... 0answers 77 views ### Proof of the cardinality of continuous functions from [0,1] to [0,1]. I've been thinking about the cardinality of continuous functions from [0,1] to [0,1]. I know that the cardinality is the same as that of [0,1] and the standard proof using the fact that such a ... 0answers 54 views ### Why does the law of cosine not work in \mathbb R^n? In class, we derived the dot product formula using the cosine law in \mathbb R^2 and \mathbb R^3. Then, we found the Cauchy-Schwarz theorem and the Triangle Inequality theorem. Then, we defined ... 1answer 17 views ### Define two operations in V={{i\in \mathbb N: i<2^n}} Let V={{i\in \mathbb N: i<2^n}}, with n fixed. Define an operation for addition and an operation for scalar multiplication so that V with those operations is a vector space over \mathbb Z_2 ... 0answers 37 views ### Variation of orthogonal vectors It is given that inner product$$ \left\langle a(t),b(t)\right\rangle =0,\quad \forall t\in[0,T]$where$a(t), b(t)\in \mathbb{R}^n$. If$\dot{a}(t)$is known, is there a way to find an expression ... 1answer 40 views ### Linear dependence of a set of vectors Is the following a correct description of a linearly dependent set of vectors: A set of vectors$S$(in a vector space$V$) is said to be$\textbf {linearly dependent}$if there exists$\textbf ...
Let's assume I have two points with coordinates $(x,y,z)$ and $(x_1,y_1,z_1)$ and there is line between them. I am given a point with coordinates $(x_2,y_2,z_2)$. What's the easiest way to calculate ...