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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630 views

dimension of the space of all symmetric matrices with trace $0$ and $a_{11}=0$,

I want to know the dimension of the space of all symmetric matrices with trace $0$ and $a_{11}=0$, I can show that the dimension of space of all symmetric matrices $S$ is $n(n+1)/2$, now I give a ...
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1answer
140 views

Product of binomial coefficient as a basis

I am stuck with the following problem. Every polynomial of degree $d$ can be expressed as $$ p(x) = p_d \binom{x}{d}+ p_{d-1}\binom{x}{d-1} + \cdots + p_0 \binom{x}{0} $$ What is the ...
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4answers
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Sphere tangent to a plane

Find the equation for a sphere with center $(\alpha,\beta,\gamma)$ tangent to the plane $ax + by + cz = d$. The sphere is $(x-\alpha)^2 + (y-\beta)^2 +(z-\gamma)^2 = r^2$ and I understand that some ...
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1answer
266 views

Span and Dimension: A subspace

If $A$ is finite set of linearly independent vectors then the dimension of the subspace spanned by $A$ is equal to the number of vectors in $A$. This is obviously true. Since $A$ is a finite set of ...
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3answers
87 views

Generating a $n$-th dimensional vector orthogonal to $n-1$ linearly-independent vectors

Let us have $n-1$ linearly independent vectors $\vec{v}_{1},\dots,\vec{v}_{n-1}\in\mathbb{R}^{n}$, define the vector $\vec{w}$ as follows: ...
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1answer
61 views

Can you define a vector space in terms of a pre-existing projective space?

Projective spaces are usually defined as the quotient of a vector space (by the equivalence relation that identifies collinear vectors). However, in my opinion, projective spaces seem intuitively less ...
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2answers
271 views

Sparse basis for linear subspace

Suppose I have a linear subspace of some vector space, e.g. described as the column space of some big matrix. How would I algorithmically find a basis of that same subspace where the basis matrix is ...
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1answer
243 views

Normal operator $f \in L(V,V)$ adjoint as a polynomial in $f, f^*=p(f)$.

I'm preparing for a Linear Algebra exam, grad school level. If $V$ is a complex vector space "unitaire" (term in French, but I can't find this term anywhere except in my class notes, I think it's ...
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1answer
60 views

Dot products of three or more vectors

Can't we construct a mapping from $V^3(R^1)$ to $R$ such that $a.b.c = a_{x}b_{x}c_{x}+a_{y}b_{y}c_{y}+a_{z}b_{z}c_{z}$ (a,b,c are vectors in $V^3(R^1)$ ) and more generally $a^n$ , $a.b.c.d.e...$ ...
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1answer
81 views

Extensions of finite-rank operators

Let $V$ be a vector space and let $W$ be its subspace of infinite codimension. Let $\mathcal{F}_W$ be the family of all finite-rank operators on $V$ with range contained in $W$. Consider the ...
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1answer
358 views

Geometry problem from a Berkeley course

I've been trying to solve this problem proposed as part of one of the first lectures of a Berkeley linear algebra course: "What Good is a Basis ? The freedom to choose a basis often simplifies ...
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1answer
346 views

Vector subspaces proof check

Suppose I wish to show that for a finite dimensional vector space, $V$, with basis $B=\{b_1,...,b_n\}$ and a given subspace $X$ of $V$, there exists a subset of $B$ that generates a subspace $Y$, such ...
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0answers
69 views

When are all ring homomorphisms also algebra homomorphisms?

Let $k$ be an algebraically closed field, and let $A,B$ be two unitary $k$-algebras. In general, there are more ring homomorphisms $A\to B$ than there are $k$-algebra homomorphisms. More precisely, ...
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0answers
112 views

Intuition about structures in Galois Theory.

Background Often in mathematics I find we can ask "why" something is true. Of course, this is not a well defined question. However, it usually prompts answers that includes words like ...
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0answers
56 views

Generating a 3d ribbon from a series of points

I am attempting to generate a 3d ribbon from a set of 3d points. The idea is to generate a realistic ribbon which follows those points. In its current state, one example looks like this: In this ...
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0answers
189 views

Are there eigenvectors, eigenvalues, and characteristic functions for non-linear vector fields?

An eigenvector is a vector in the preimage of the transformation whose direction is not changed when the transformation is applied. It seems like the concept of eigenvectors and eigenvalues would ...
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1answer
189 views

Cross Product Intuition

I know the cross product between a vector $a$ and a vector $b$ is just a vector whose magnitude is the product of magnitude of $b$ times the magnitude of the perpendicular component of $a$ in relation ...
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0answers
128 views

Why is the radical of a Clifford algebra generated by the kernel of the associated symmetric form? [duplicate]

I was recently reading through Jacobson's Basic Algebra. I got to the section on Clifford algebras, and have the following question. Let $Cl_\omega$ be the Clifford algebra with bilinear symmetric ...
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5answers
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What is the difference between metric spaces and vector spaces?

Does a metric space have an origin? That is, does it have $(0,0)$. Does a vector space have an origin? It seems whatever you can do in a metric space can also be done in a vector space. Is this ...
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2answers
401 views

Dimension of $\mathbb{Q}\otimes_{\mathbb{Z}} \mathbb{Q}$ as a vector space over $\mathbb{Q}$

The following problem was subject of examination that was taken place in June. The document is here. Problem 1 states: The tensor product $\mathbb{Q}\otimes_{\mathbb Z}\mathbb{Q}$ is a vector ...
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4answers
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What are some alternative definitions of vector addition and scalar multiplication?

While teaching the concept of vector spaces, my professor mentioned that addition and multiplication aren't necessarily what we normally call addition and multiplication, but any other function that ...
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3answers
383 views

Shortest length that a vector can have

I came to the following question from a past exam: The vector $v = (k, k, 3 − k)$ depends on a variable $k$. What is the shortest length of the vector $v$ can have? I know that the answer is ...
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7answers
30k views

Finding a unit vector perpendicular to another vector

For example we have the vector $8i + 4j - 6k$, how can we find a unit vector perpendicular to this vector?
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3answers
257 views

If $Ax=B$ has two solution, then there must be a third one?

How do I prove this conjecture? Let $A$ be a matrix, and $B$ be a column vectore. If $Ax=B$ has two solutions, then there must be a third one. Thanks in a advance!
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1answer
223 views

Is it possible to construct a quasi-vectorial space without an identity element?

I mean if there is any construction that satisfies all the conditions for an vectorial space except it lacks an identity element? This questions was posed to me by a classmate last semester and I have ...
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6answers
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Prove in full detail that the set is a vector space

So I'm doing a review test and I have this problem: Prove in full detail, with the standard operations in R2, that the set {(x,2x): x is a real number} is a ...
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3answers
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Vector Spaces: Finding a basis and Dimension

I could really use some step-by-step help on these two problems please. Thank You in advance. 1.) Let $V = \{{\bf{A|A}}$ is an $n \times n$ matrix, $n$ fixed, det$({\bf{A}}) = 0$ }. Is $V$, with ...
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3answers
197 views

Embedding torsion-free abelian groups into $\mathbb Q^n$?

Glass' Partially Ordered Groups states without proof: Every torsion-free abelian group can be embedded into a rational vector space (as a group). Can someone link me to a proof of this? It ...
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3answers
101 views

For subspaces, if $N\subseteq M_1\cup\cdots\cup M_k$, then $N\subseteq M_i$ for some $i$?

I have a vector space $V$ over a field of characteristic $0$. If $M_1,\dots,M_k$ are proper subspaces of $V$, and $N$ is a subspace of $V$ such that $N\subseteq M_1\cup\cdots\cup M_k$, how can you ...
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2answers
363 views

Meaning of, and how to verify, a vector space *over* $\mathbb{R}$

In Axler's book on Linear Algebra he writes ($\mathbb{F}$ is here either $\mathbb{R}$ or $\mathbb{C}$): The scalar multiplication in a vector space depends upon $\mathbb{F}$. Thus when we need to ...
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2answers
205 views

The isomorphisms between two vector spaces

Let $V$ and $W$ be two vector spaces over real number field, if they are isomorphic as vector spaces over rational number field, are they isomorphic as real vector spaces ?
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How do you prove that tr(B^(T) A ) is a inner product?

Consider the vectorspace of all real $m \times n$ vectors and define an inner product $\langle A,B\rangle = \operatorname{tr}(B^T A)$. "tr" stands for "trace" which is the sum of the ...
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1answer
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Proof for triangle inequality for vectors

Generally,the length of the sum of two vectors is not equal to the sum of their lengths. To see this consider the vectors $u$ and $v$ as shown below. By considering $u$ and $v$ as two sides of a ...
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1answer
155 views

Is $\mathbb{R}^1$ a subspace of $\mathbb{R}^2?$

My intuition tells me it is. But in terms of vectors, the span of a vector with only one component (a vector in $\mathbb{R}^1$) is not said to be a subspace of $\mathbb{R}^2$
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2answers
455 views

If a field $F$ is such that $\left|F\right|>n-1$ why is $V$ a vector space over $F$ not equal to the union of proper subspaces of $V$

If $U_1$, $U_2,\ldots,U_n$ are proper subspaces of a vector space $V$ over a field $F$, and $|F|\gt n-1$, why is $V$ not equal to the union of the subspaces $U_1$, $U_2,\ldots,U_n$?
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5answers
490 views

Help a newbie understand Linear Algebraic terms

I am taking a class in Algebra but I am having a problem grasping exactly what it is I am being asked to do -- I think I am having a problem with the vocabulary being used. I have a couple of ...
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5answers
174 views

Proof involving subspaces

I encountered this question in a document I found on a google search, it bugged me because my perception keeps telling me I'm wrong no matter what I do. Let $U$, $W$ and $Z$ be subspaces of a ...
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3answers
105 views

Prove that $0 < x < y$ implies $\|x\| < \|y\|$ for any norm.

All vectors are real. Prove that $0 < x < y$ (element-wise) implies $\|x\| < \|y\|$ for any norm. This is probably very basic, but I don't seem to get the hang of it. Edit: it turns out this ...
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1answer
391 views

What is the kernel of the tensor product of two maps?

Assume that $f_1\colon V_1\to W_1, f_2\colon V_2\to W_2$ are $k$-linear maps between $k$-vector spaces (over the same field $k$, but the dimension may be infinity). Then the tensor product ...
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2answers
124 views

Is an infinite linearly independent subset of $\Bbb R$ dense?

Suppose $(a_n)$ is a real sequence and $A:=\{a_n \mid n\in \Bbb N \}$ has an infinite linearly independent subset (with respect to field $\Bbb Q$). Is $A$ dense in $\Bbb R?$
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4answers
345 views

Can a vector space have multiple spanning sets?

Maybe this is obvious, but can a vector space have multiple spanning sets or is there only a single spanning set for every vector space? Thanks
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1answer
222 views

How to show that two vector spaces $V$ and $W$ are the same

How to show that two vector spaces $V$ and $W$ are the same, if we know $\dim V = \dim W$ and $V$ is a subspace of $W$ ? Would it suffice to show there exists an isomorphism between them ? Any help ...
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3answers
482 views

Does $\det(A) \neq 0$ (where A is the coefficient matrix) $\rightarrow$ a basis in vector spaces other than $R^{n}$?

I know that for a set of vectors $\{ v_{1}, v_{2}, \ldots , v_{n} \} \in \mathbb{R}^{n}$ we can show that the vectors form a basis in $\mathbb{R}^{n}$ if we show that the coefficient matrix $A$ has ...
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2answers
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Realification and Complexification of vector spaces

I am interested in a good comprehensive resource on realification and complexification of vector spaces over the reals or complexes (and the interplay of these structures on the 'same' space in ...
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2answers
474 views

Is $(\mathbf{V} \cap \mathbf{W})^{\bot}=(\mathbf{V}^{\bot} \cap \mathbf{W}^{\bot})$?

Is $(\mathbf{V} \cap \mathbf{W})^{\bot}=(\mathbf{V}^{\bot} \cap \mathbf{W}^{\bot})$? I tried element-chasing, but I am getting confused when trying to determine mutual containment.
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2answers
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How do you calculate the unit vector between two points?

I'm reading a paper on fluid dynamics and it references a unit vector between two particles i and j. I'm not clear what it means by a unit vector in this instance. How do I calculate the unit vector ...
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votes
3answers
175 views

Is it possible to swap vectors into a basis to get a new basis?

Let $V$ be a vector space in $\mathbb{R}^3$. Assume we have a basis, $B = (b_1, b_2, b_3)$, that spans $V$. Now choose some $v \in V$ such that $v \ne 0$. Is is always possible to swap $v$ with a ...
4
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1answer
195 views

Normal $T\in B(H)$ has a nontrivial invariant subspace

I am wondering if the following is true: Every normal $T\in B(H)$ has a nontrivial invariant subspace if $\dim(H)>1$?
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2answers
386 views

A question about inner products on abstract vector spaces

I have been reading some materials and, for the n-th time in my life, there was a definition of an inner product as a function $V \times V \rightarrow F$, where $V$ is an abstract vector space and $F$ ...
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2answers
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Convert angle (radians) to a heading vector?

I have been looking everywhere trying to find out how to convert an angle in radians (expressed as -Pi to Pi) to a heading vector. The only [x,y] answer I have found is, [cos(angle), sin(angle)] , ...