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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36 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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66 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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1answer
70 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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86 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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245 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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386 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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100 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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1answer
38 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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94 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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35 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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32 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
20 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
47 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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82 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
46 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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40 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: ...
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1answer
41 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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1answer
203 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 $y$ ...
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1answer
176 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
144 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
39 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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1answer
339 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
142 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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37 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 ...
2
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2answers
71 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
110 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 ...
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1answer
129 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
44 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 ...
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2answers
39 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
91 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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36 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
33 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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43 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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411 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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1answer
874 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 ...
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1answer
105 views

Difference between lines dividing planes and planes dividing space

Let a(n) represent the number of regions that the plane R2 is broken into by n lines (no 2 of which are parallel, and no 3 of which intersect in a single point). Let b(n) represent the number of ...
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56 views

Gradient function

Let A (red) and B (green) 2 distinct points anywhere in a 3D space. I am looking for a function which take a point P, and returns the value in blue in the picture. Each blue number in the picture ...
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1answer
55 views

dimension of an intersection of subspaces

Let $V$ be the vector space of all polynomials in one variable with real coefficients having degree at most 20. Define the subspaces \begin{align*} W_1 &=\{p \in V; p(1)=0,p(1/2)=0, ...
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1answer
290 views

Vector spaces and multiplicative inverse?

Do vector spaces have multiplicative inverses? They seem to be monoids under $+,\times$, so monoids $(\Bbb F, +)$ and $(\Bbb F, \times)$ where $\Bbb F=\Bbb R \,or\, \Bbb C$ And it is even a group ...
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1answer
55 views

How to find the normal vector in a TNB problem

I have done this TNB problem multiple times; however, my online homework system keeps telling me my answer is incorrect. I was hoping someone would look at my work and tell me where I'm going wrong? ...
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1answer
78 views

Finding a linear transformation such that $T^{3} = T $

I have to show that there exists a linear transformation such that $T^{3} = T $ i can see that from here that T has eigen values $0.1.-1$ .But how do i find linear transformation .Also for v and q ...
2
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1answer
36 views

Length of a complex vector

From the definition of inner product in $\mathbb{F}^n$ $$\textbf{a}\cdot\textbf{a}=\sum\limits_{k=1}^na_{k}\overline{a_{k}}$$ Say $a_{k}=x_{k}+iy_{k}$, then ...
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1answer
126 views

how can I calculate the 4 corners of a finite plane that rests in a 3d space

I have a finite plane in my application. The plane is described by its centre point C, its normal vector N and a scale vector S. S is not really a vector but rather a "convenient container" of scale ...
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335 views

Relation between the vector space of Fibonacci sequences and $\mathbb{R}^2$

Question: Let $V$ be the vector space of real sequences over $\mathbb{R}$. If $W$ is the subspace of all Fibonacci sequences (i.e. a sequence $\{a_n\}\in W$ if $a_n=a_{n-1}+a_{n-2}$, for all $n\geq ...
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1answer
72 views

Misunderstanding in the proof that the sum of subspaces is the smallest containing subspace.

So if $V_1,...,V_n$ are subspaces of $M$ then $V_1+...+V_n$ is the smallest subspace of $M$ containing $V_1,...,V_n$ The proof is that clearly $V_1,...,V_n$ are all contained in $V_1+...+V_n$ Then ...
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1answer
35 views

Two sets of polynomials with distinct roots build the ring of polynomials.

Definitions: $i \in K$ $U_{i}:=\{f\in K[X] |f(i)=0 \}$ $K[X]$ is the ring of polynomials HINTS: K[X] is a vector space Every $U_{i}$ is a vector subspace of $K[X]$ Question: (i) With $s \neq ...
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1answer
293 views

Linear Transformations on Infinite Dimensional Vector Spaces

Let $T$ be a linear transformation $T:V\to V$, where $V$ is an infinite dimensional vector space. How can we construct examples such as $1.$ T is one to one but not onto $2.$ T is onto but not ...
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1answer
396 views

The set of differentiable functions such that $f'(2)=b$ is a linear subspace if and only if $b=0$??

Questions are in bold. The set of differentiable real-valued functions on (0,3) such that $f'(2)=b$ is a subspace of $(0,3)\to \mathbb R$ if and only if $b=0$ ($(0,3)\to \mathbb R$ denotes the set of ...
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2answers
334 views

Matrix notation of vectors?

My linear algebra book says that a vector is a one-column matrix. However, in the context of what we are studying (linear equations) it would make more sense if a vector was of the form of the ...
2
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2answers
51 views

Dimension of $n \times n$ diagonal matrix with characteristic polynomial $(x-a)^{p}(x-b)^{ q}$?

What is the dimension of an $n \times n$ diagonal matrix with characteristic polynomial $(x-a)^{p}(x-b)^{ q}$? Do I have to make distinct cases with as $p + q < n$ and equal to $n$? And if their ...