For questions about vector spaces and their properties. More general questions about linear algebra belong under the [tag:linear-algebra] tag. A vector space is a space which consists of elements called "vectors", which can be added and multiplied by scalars. In other words, these are the spaces ...

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Change of basis matrix for polynomials

I've already asked this question here, but theres a misconception about the phrase "change of basis matrix from B to C", and I think the answers were given in the inverse of what's in my book. So, in ...
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3 views

If $f ' (x ; y)=0$ for every $x$ in open convex set, then $f$ is constant on open convex set.

$f′(x;y)=0$ for every $x$ in an open convex set $S$ and every $y$ in $\mathbb{R}^n$, Prove that $f$ is constant on $S$. $f′(x;y)$ is the derivative at $x$ in the direction $y$. Seems like I have ...
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1answer
8 views

Separating family of seminorms covers a vector space

Let $\mathcal{P}$ be a separating family of seminorms on a vector space $X$. Show that if $x\in X-\{0\}$ then $\exists p\in\mathcal{P}$ such that $p(x)\leq1$. Context: This is from theorem 1.37 in ...
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1answer
30 views

Verify if this set of matrices span $M_2(\mathbb{R})$

I have the set: $$S = \left\{\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}, \begin{bmatrix}1 & 1\\0 & 0\end{bmatrix}, \begin{bmatrix}0 & 0\\1 & 1\end{bmatrix}, \begin{bmatrix}0 ...
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8 views

Finding coordinate axis vectors to yield a specific transformation

For a project I am working on, I will be given a point $\textbf{a}$ in a global 3-D coordinate system, as well as a point $\textbf{a}^\prime$ in a local coordinate system. (The global coordinate ...
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13 views

Let $F$ and $G$ be linear transformations. Find the transformation matrix in respect to basis $B$ and $C$. (polynomials)

Let $F$ and $G$ be two linear transformations that maps from $P_2(\mathbb{R})$ to $P_3(\mathbb{R})$, such that: $$F(p(t)) = tp(t)-p(1)\\G(p(t)) = (t-1)p(t)$$ Find the transformation matrices of $F$ ...
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1answer
24 views

Proving statements , that require Zorn's lemma , for countable case directly by well-ordering principle of natural numbers

We know that for countable sets , the existence of a choice function is a consequence of the well-ordering principle ; and it is also known that the results like "every vector space has a maximal ...
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2answers
35 views

How can I parametize a curve of intersection of two surfaces?

To find out directional derivative $f(x.y.z)=x^2+y^2−z^2$ at $(3,4,5)$ along the curve of intersection of the two surfaces $2x^2+2y^2−z^2=25$ and $x^2+y^2=z^2$ I am trying to parametrize above two ...
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1answer
13 views

Cartesian equations for the line tangent to two surface.

I am asked to find a Cartesian equation for the line tangent to both the surfaces x^2+y^2+2z^2=4 and z=e^(x-y) at the point (1.1.1) I tried to find out normal vector to both surfaces and tangent ...
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2answers
18 views

Find Point on the line segment (7/8) of the way connecting points P and Q

with P = (4,3,-4) and Q = (5,-4,3). My thinking is take the distance between the two, which is (1,-7,7) and taking 7/8 of it which is (-7/8,-49/8,49/8). But I feel like that is wrong and I have to ...
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1answer
55 views

References for hemicontinuity?

Let $X$ be a real vector space, $K\subset X$ be a nonempty and convex set. The mapping $f:X\rightarrow\mathbb{R}$ is said to be hemicontinuous if for every $u,v\in K$, the mapping ...
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0answers
10 views

How would you find out the cartesian equation tangent to two surfaces at given point?

I am given two surfaces and asked to find out a pair cartesian equations that are tangent to these two surfaces. I know how to find out tangent plane to one surface. But, how would you find out a ...
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1answer
31 views

For which values ${(a,1,0),(1,a,1), (0,1,a)}$ form a basis?

To the set be a basis we should have: $$x(a,1,0)+y(1,a,1)+z(0,1,a) = (0,0,0)\implies x=y=z=0$$ so: $$ax + y = 0\\x + ay + z = 0\\y + az = 0$$ which is a system that only has a unique solution if ...
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1answer
24 views

Evaluate the directional derivative along the curve of intersection of the two spheres..

I am given $f(x.y.z)=x^2+y^2-z^2$ at $(3.4.5)$ along the curve of intersection of the two surfaces $2x^2+2y^2-z^2=25$ and $x^2+y^2=z^2$ And evaluate the directional derivative. I know how to find ...
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0answers
22 views

W subspace of $\mathbb{R^4}$ with $\mbox{dim}W = 3$ and $U =\cdots $. Find $\mbox{dim} U+W$ and $\mbox{dim} U\cap W$

Let $W$ be a subspace of $\mathbb{R^4}$ with $\mbox{dim}W = 3$ and $U = \mbox{span}((1,2,1,3),(3,1,-1,4))$. Find $\mbox{dim} U+W$ and $\mbox{dim} U\cap W$ Well, so $\mbox{dim } U = 2$, clearly, ...
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0answers
46 views

I need help in Algebra. [on hold]

can someone explain to me the Vector space and the Vector base? (their uses and how to find a matrix' base and space and so on..) thanks in advance
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1answer
20 views

Vectors components that are not contra or covariant?

I know that a vector can have contravariant and covariant components, but is it possible to have components that are neither contravarient or covariant? I suspect that the answer is yes, and that most ...
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1answer
17 views

If $∇f(a)\cdot y ≤ 0$ for every vector $y$, why does $\nabla f(a)$ have to be zero?

If $f$ is differentiable at every point in $B(a)$ and $f(x)≤f(a)$ for all $x$ in $B(a)$, prove that $∇f(a)=0$. I actually did some work and found out that $∇f(a)\cdot y ≤ 0$ for every vector $y$. ...
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1answer
25 views

Assume $f$ is differentiable at every point of $B(a)$ and $f(x)$ is less than or equal to $f(a)$

Over the scalar field, If $f$ is differentiable at every point in $B(a)$ and $f(x)$ is less than or equal to $f(a)$, prove why gradient of $f(a)$ is $0$. Just don't understand how to start with,
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1answer
19 views

Derivative over scalar field with respect to fixed point proof.

Prove there is no such scalar field that $f '(a;y) >0$ for fixed point $a$ and every non-zero vector $y$. I posted this question but some of you pointed out that it is not clear. So, $f ' (a;y)$ ...
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2answers
32 views

Proving linear dependency for two vector groups

The question: Let V be a vector space over $\mathbb{R}$. Let $S = \{v,u,w\}$ be a group of 3 vectors in V. Let T be defined as $T = \{v, v + u, v + u + 2w \}$. Prove that if S is linearly dependent, ...
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0answers
16 views

See if vector set is basis of space using Gram Schmidt process

I have a problem my teacher gave me and I can't find an answer. She gave me a set of 3 vectors, $$\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} \begin{bmatrix} 7\\3\\1 \end{bmatrix} \begin{bmatrix} ...
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2answers
20 views

Let $V$ be a finite dimensional linear space and let $S$ be a subspace of $V$. Prove that a basis for $V$ need not contain a basis for $S$.

Let $V$ be a finite dimensional linear space and let $S$ be a subspace of $V$. Prove that every basis of $S$ is part of a basis for $V$ but a basis for $V$ need not contain a basis for $S$. Attempt: ...
2
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1answer
29 views

Finding linear independence in $v_1,\ldots,v_m$

First, I'll try not to ramble, although it tends to happen when I type. I have the following linear algebra problem for my homework. Prove or give a counterexample: If $v_1, v_2, \ldots , v_m$ are ...
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7 views

mean value theorem for scalar field

I just want to make sure that Mean value theorem for scalar field works same as one- dimensional mean value theorem. Usually, my book explains Mean value theorem for scalar field on interval [0.1]. ...
2
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0answers
6 views

Mean value theorem and scalar field proof

Assume that f′(x;y)=0 for every x in some n-ball B(a) and for every vector y. Use the mean value theorem to prove that f is constant on B(a). And if f′(x;y)=0 for a fixed vector y and for every x in ...
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0answers
12 views

Can anyone help me prove derivative of scalar field using mean value theorem?

Assume that f′(x;y)=0 for every x in some n-ball B(a) and for every vector y. Use the mean value theorem to prove that f is constant on B(a). And if f′(x;y)=0 for a fixed vector y and for every x ...
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0answers
22 views

There is no scalar field such that $f '(a)>0$ for fixed $a$ and for every nonnegative vector $y$ [on hold]

I am trying to prove this. But can't think of how I should start. Anyone has some ideas? and why is there a scalar field $f'(a)>0$ for every $a$ and for fixed vector $y$ ? can anyone give me an ...
2
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1answer
27 views

Trying to understand proof that 3 non-collinear points determine a unique plane

$Q,R,P$ are 3 non-collinear points. Plane $M = P + s(Q-P) + t(R-P)$. Let $C = Q-P$ and $D= R-P$. Let us grant that C and D are linearly independent. Let $M' = P + sA + tB$. Assume $M'$ has $P,Q,R$. ...
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1answer
25 views

Help to understand the basis for a dual space

I've been introduced to the concept of dual space in linear algebra. I can understand perfectly that the dual space of the space $V$ is a space $V^*$ made of all possible linear maps from $V$ to ...
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1answer
33 views

Can there be ever a counterexample to this?

Does addition on subspaces have an additive identity? I said yes because subspaces are vector spaces, so they must have an additive identity. Which subspaces have additive inverses? I said all of ...
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5 views

Finding rotation axis and angle to align two 3D vector bases

I have asked this question before and, while the accepted answer solved my problem back then, I am still interested in finding the rotation axis and angle. Let me rephrase the problem here: I would ...
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1answer
17 views

Field extension of a vector space

If $V$ is a vector space over the field $k$, and $K$ is a field extension of $k$, then $(V)_K$ over $K$ is a vector space. How this new vector space is constructed? and how are the linear ...
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0answers
29 views

Prove that $U_1\cup U_2$ is a subspace of $V$ $\iff$ $U_1\subseteq U_2$ or $U_2\subseteq U_1$ $\triangle$

Let $V$ be a vector space over some field. Let $U_1$ be a subspace of $V$. Let $U_2$ be a subspace of $V$. Prove that $U_1\cup U_2$ is a subspace of $V$ is equivalent to $U_1\subseteq U_2$ or ...
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0answers
12 views

Dimension of sum of permutations of tensor products of vector spaces

Sorry for the mouthful of a title! Suppose I have two finite vector spaces $W,V$ with bases $\{w_1\dots w_p\}$ and $\{v_1\dots v_q\}$. Consider some subspace $S$ of $W\otimes V$ of dimension $m$ ...
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0answers
16 views

How to understand the meaning of 'Oblivious' in Oblivious Subspace Embedding?

For the definitions of Oblivious Subspace Embedding and Subspace Embedding, please refer to the 1st page of paper http://arxiv.org/pdf/1308.3280v1.pdf.
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2answers
49 views

Lattice of integers $\mathbf{Z}$ in $\mathbb{R^2}$

Lattice of integers $\mathbf{Z}$ in $\mathbb{R^2}$ The questions: Give an example of a nonempty subset of $\mathbb{R^2}$ (noted $M$) which is closed under addition and for all $m\in M$ we have $-m\in ...
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1answer
70 views

Under which branch of mathematics do vectors fall into? [on hold]

Well... this is pretty basic. Under which branch of mathematics do vectors fall into? A quick search on the web revealed many kinds of vectors so what I have in mind are this kind of vectors: that ...
3
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1answer
39 views

Verifying a Vector Space Via Given Axioms

Let $X$ be the collection of all sequences $\{\alpha_n\}_{n=1}^{\infty}$ of scalars from $\mathbb{K}$ such that $\alpha_n=0$ for all but a finite number of values of $n$. Define addition and scalar ...
2
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1answer
91 views

Linear Transformation on $\mathbb{R}^6$

Let $W$ be a vector space over $\mathbb R$ and let $T:\mathbb R^6 \to W$ be a linear transformation such that $S = \{Te_2, Te_4, Te_6\}$ spans $W$. Wich one of the following must be true? ...
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1answer
20 views

Vector Spaces and Simple Modules

Let $G$ be a finite group and let $R = \textbf{R}[G]$ be the group ring of $G$ with coefficients in the field $\textbf{R}$ of real numbers. Let $V$ be an $R$-module which is finite-dimensional as an ...
2
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2answers
23 views

Linear transformation with special properties

how should I do that please (I had this in my test yesterday)? Linear transformation $f:\mathbf{R}^{10} \to \mathbf{R}^7$ has an attribute that every vector $\mathbf{v}$ for which is true that ...
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2answers
80 views

Is $\mathbb{R}^2$ the same as my dear $\mathbb{C}^2$?

The question is$$\text{Is }\mathbb{R}^2\text{ a subspace of }\mathbb{C}^2?$$My first thing to think about it now is $$\text{Is }\mathbb{R}^2\text{ a subset of }\mathbb{C}^2?$$ I think no because what ...
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0answers
29 views

Try to prove linear independence

I know that since vectors $v$ and $w$ are linearly independent, $av+bw=0$. Should I continue with the assumption that $v, w, v \times v$ are linearly independent so get $av+bw+c(v \times w)=0$? If ...
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1answer
16 views

Span of a set of vectors

In Artin's Algebra book there is the following Lemma about vector spaces: Let $S$ be an ordered set of vectors of $V$, and let $W$ be a subspace of $V$. if $S\subset W$, then Span $S\subset W$. Now, ...
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1answer
16 views

Proof that the velocity vector is tangential to the path?

In calculus class my teacher asserted that the velocity vector is tangential to the path a point takes. I have tried to prove this but have gotten stuck. I computed $\dfrac{v_y}{v_x}$ to be ...
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1answer
24 views

Is every finite dimensional linear space a banach space [on hold]

Is every finite dimensional linear space a Banach Space? Is every finite dimensional linear space a Hilbert Space?
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0answers
24 views

How do I find the common invariant subspaces of a span of matrices?

Let $G_1, \ldots, G_n$ be a set of $m\times m$ linearly-independent complex matrices. Let $\mathcal{G} = \operatorname{span}\left\{ G_1, \ldots , G_n\right\}$ be the vector space that spans the set ...
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0answers
38 views

prove the subspace is equal to L1∩L2

Let $L1,L2$ be subspaces of $V$ and $dim(L1+L2)=1+L1 \cap L2$ Show that $L1 \subseteq L2$ or $L2 \subseteq L1$ and $L1+L2= L1$ or $L2$ it's has something to do with if $L1,L2 \ne L1 \cap L2$ ...
1
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1answer
22 views

Cosine Similarity between two sets of vectors?

I have words represented as vectors, and so I can compare two words using the cosine similarity of each word vector. But, now I'd like to extrapolate that and compare two sentences, each being a set ...