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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Equation of plane that goes for intersection of 2 planes and is perpindicular to another plane

Really don't know what to do here, went to a tutor neither did he. Okay first the problem: Find the equation of the plane that passes through the line of intersection of the planes x − z = 2 and y + ...
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4answers
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Orthogonal transformations and cross product

In the appendix of Pressley's Differential Geometry, he states: For a $3\times 3$ orthogonal matrix $M$, for $u,v\in \mathbb{R}^3$, $Mu\times Mv=(\det M)(u\times v)$, i.e. $Mu\times Mv=\pm u\times v$ ...
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1answer
73 views

How much linearly independent? or linearly dependent?

I want to improve a rank-deficient matrix by augmenting a row vector to it. However, unfortunately, I have only very 'similar' vectors.. For example, my matrix is somewhat like.. \begin{bmatrix} 1 ...
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1answer
61 views

Vector spaces with unique basis

Vector spaces like $\mathbb{R}^n$ can have different bases and we can change the basis with a matrix to get a new one. This made me wonder: Are there any vector spaces with $dim>1$ that have only ...
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1answer
105 views

elementary abelian groups and finite fields

in group theory, an elementary abelian group is a finite abelian group, where every nontrivial element has order $p$, where $p$ is a prime; it is a particular kind of $p$-group. now suppose that we ...
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218 views

Computing Euler Angles from Direction Cosines Vector

My problem simply as the following: Suppose we measured the orientation of a plane object with respect to a reference fame. (where the reference frame parallel to plane frame). The unit normal vector ...
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1answer
264 views

Prove that set is orthonormal set

In vector space of all real polynomials with inner product $(x,y) = \sum_0^1x(t)y(t)dt$. $x_n(t) = t^n$ for $n = 0, 1, \dots $. Show that functions: $y_0(t) = 1, $ $y_1(t) = \sqrt(3)(2t-1), $ ...
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linear independence and reduced row echelon form

If I can write vectors $(2,0,0)$ ,$ (1,-1,0)$ and $(0,1,1) $ as $\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1 \end{bmatrix}$ using reduced row echelon form does this means that they ...
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1answer
48 views

Find isomorphic vector spaces $V$ and $W$ with $V = S \bigoplus B$ and $W = S \bigoplus D$ but $B \ncong D$.

I'm having some trouble with this. It seems very difficult to construct the spaces in such a way that $ V = S \bigoplus B$ and $W = S \bigoplus D$ with the conditions that $ B \ncong D$. Any help ...
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2answers
304 views

If V is an infinite-dimensional vector space, and S is an infinite-dimensional subspace of V, must the dimension of V/S be finite? Explain

I'm having some issues thinking about this one. I'm going to think aloud here. If $V$ is infinite dimensional, this means that there infinitely many basis vectors. I know the choice of the ...
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1answer
68 views

automorphism of fields

let we consider $GF(p^n)$ as a vector space over $GF(p)$, $p$ is prime. Also we want to have an invertible linear map on $GF(p^n)$, (automorphism of $GF(p^n)$). on the other hand we know that A field ...
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2answers
45 views

Is vectorspace trivial under these conditions?

Let $R$ be a ring. Looking for a left $R$-module free over abelian group $A$, I arrived at $\left|R\right|\otimes A$ with $r.\left(s\otimes a\right)=rs\otimes a$ where $\left|R\right|$ denotes the ...
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2answers
61 views

Proving an inequality with the Schwarz inequality

Given a vector space with a Hermitian dot product defined, prove the following inequality using the Schwarz inequality. Let $f$ be a complex value function that is continuous within $0 \le x \le 1$, ...
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6answers
325 views

Which are the most effective modern intuitive definitions of a vector?

First, I would like to clarify what I mean by "intuitive definition": an intuitive definition is an informal understanding of a concept which helps to build mental agility, with the possible ...
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1answer
103 views

Multivariable Calculus Vector Fields

I have to prove that if $f(x,y,z)=f_{a}(x,y,z)+f_b(x,y,z)+f_c(x,y,z)$ is a conservative vector field and and $g(x,y,z)=g_{a}(x,y,z)+g_b(x,y,z)+g_c(x,y,z)$ is also a conservative vector field, then ...
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1answer
280 views

How to calculate the points of the triangles making up an Octahedron?

Ok guys, I'm not a great mathematician but will try to work this as accurately as I can. I hope someone can help me. I am drawing some 3D objects and I am having trouble drawing an Octahedron. I ...
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1answer
40 views

dimension of a subspace

Suppose that x and y are vectors and M is a subspace in a vector space V. Let H be the subspace spanned by M and x. Let K be the subspace spanned by M and y. Prove that if y is in H but not in M , ...
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1answer
57 views

Vector space definitions, difficulties

First, these were homework questions. However, having already submitted them, asking this doesn't break any code. This is a new definition of vector addition in $\mathbb{R}^3$: $$(x_1, y_1, z_1) ...
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3answers
36 views

Find the unit normal $N$

of ${\bf r}=14 \mathrm{e}^{-10 t}\cos(t){\bf i}+14 \mathrm{e}^{-10 t}\sin(t){\bf j}$ The answer should be in vector form. I can't find an easy way to do this. I end up with T being something ...
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2answers
462 views

Subspaces of $\mathbb R^3$?

Which of the following sets are subspaces of R^3? A. $\{ (x,y,z)\mid x + y + z = 0 \}$ B. $\{ (x,y,z) \mid x < y < z \} $ C. $\{ (7 x - 5 y, 2 x - 6 y, 9 x + 6 y ) \mid x,y \text{ ...
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1answer
71 views

Is it always true that a subspace of a vector space is in direct sum with its orthogonal?

Let $V$ be a finite-dimensional vector space of dimension $n$ over the field $K$; and let $g:V\times V\to K$ be a non-degenerate scalar product (by ''scalar product'' I mean a symmetric bilinear map, ...
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2answers
65 views

Polynomial vector space terminology

Consider the vector space $P$ and the subset $V$ of $P$ consisting of those vectors (polynomials) $x$ for which a) $2x(0) = x(1)$, b) $x(t) = x (1-t)$ for all $t$. In which of these cases is $V$ a ...
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Problem, angle between vectors

I am trying to calculate the angle between two vectors. As I understand, the dot product of two vectors is equal to the angle. What I cant grasp is this: Given a vector $A$ and $B$ where $A = ...
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1answer
90 views

Show that all $f$ integrable on $[0,1]$ with $\int_0^1 f(x)dx = 0$ is a vector space

Let $V =$ the set of $~f$s integrable on $[0,1]$ with $\int_0^1 f(x)dx = 0$. I want to show that it is closed under addition. Let $f$ and $g$ be in $V$. $f + g$ be in $V$. Then since the integral of ...
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3answers
188 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 ...
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2answers
224 views

Show that set subset of polynomials is a subspace, and show dimension of S

Let P be the set of polynomials of degree $\leq n$, where $n$ is fixed. Let S be a subset of P s.t. the polynomials in S satisfy f'(0)= 0 and f(0) = 0. (I'm assuming this means that this set is ...
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3answers
167 views

Are there any examples of vector spaces over non-numerical fields? If not, why not?

By non-numerical vector spaces I mean vector spaces that do not have as their scalars some sort of easily discernible numerical fields (e.g. complex numbers, functions are usually maps from one ...
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2answers
377 views

union of two linear independent set

I have two linearly independent sets in $R^n$ space.One with cardinality $n-k$ and another with cardinality $k+1$.Is the union of these two sets linearly independent?Actually I came across this ...
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62 views

$x,y$ are linearly depending iff $|\langle x,y\rangle|=||x|| \cdot ||y||$

I tried to prove a special case of Cauchy-Schwarz: $$x,y \text{ are linearly depending vectors} \Leftrightarrow |\langle x,y\rangle|=||x|| \cdot ||y||$$ $\Rightarrow$ is simple: \begin{eqnarray*} x= ...
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1answer
75 views

How to distinguish vector space from $k$-algebra?

Let $A$ be a finitely generated $k$-algebra with no nilpotents. What do I need to show in order to prove it's a finite dimensional vector space over $k$? For example, is it enough to show that ...
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2answers
437 views

Finding a line that is orthogonal to other 2 lines and intersects them?

The lines given below are skew. Find parametric equations for the line which intersects them both orthogonally. $l1 : x = 6 + 2t; y = -1 + t; z = -5 - 2t$ $l2 : x = 8 - 5t; y = -1 + 2t; z = 3 - t$ ...
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2answers
87 views

How to find the shortest distance from a line to a solid?

The equation $x^2 + y^2 + z^2 - 2x + 6y - 4z + 5 = 0$ describes a sphere. Exactly how close does the line given by $x = -1+t; y = -3-2t; z = 10+7t$ get to this sphere? So the sphere is centered at ...
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3answers
231 views

Finite-dimensional vector spaces and vector components…

Are the following remarks correct ? We've got the vector space $\mathbb{R^n}$ (for the following it doesn't matter whether we identify it with $M_{1 \times n}$ or $M_{n \times 1}$). We define $e_1, ...
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2answers
110 views

Unions of subspaces

So I need to prove that for the union of $n$ subspaces to be a subspace, each subspace must be a subset of another one of the subspaces. My thought process so far is that I need to prove that it is ...
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2answers
89 views

What does the term “distinguished basis” mean?

I know what a basis is (talking about vector spaces here), but I don't know what a distinguished basis is. Can you please explain the difference to me? I did not grow up in an English-speaking ...
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2answers
171 views

Polynomial paths

I am studying Vector Space and came across the following problem in Artin: If $x(t)$ and $y(t)$ are quadratic polynomials with real coefficient, show that the image of the path $(x(t), y(t))$ is ...
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0answers
46 views

Motivation behind Definition of Projection [Poole P27]

In the long paragraph above equation $(2)$, http://mathinsight.org/dot_product avers: This leads to the definition that the dot product $\mathbf{a⋅b}$, divided by $∥\mathbf{b}∥$ (= magntitude of ...
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1answer
249 views

Most “beautiful” presentations of the basic proofs for vector spaces?

I am familiar with the standard proofs presented in textbooks for stuff like linear independence/dependence, the dimensions of common vector spaces, any basis for a vector space V must be linearly ...
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1answer
306 views

Proving that every (possibly infinite) span contains a basis in a vector space.

The book "A First Course in Algebra" says In a finite dimensional vector space, every finite set of vectors spanning the space contains a subset that is a basis. All that is fine. But what about ...
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1answer
99 views

Linear Algebra Vector Polynomial and linear independence

I was just wondering if the dimension of P4. Does that mean that all the vector space of all polynomials of degree less than or equal to four, has the dimension 4? Also if a set of vectors ...
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1answer
236 views

Basis for the subspace w spanned by {v1,v2,v3,v4}.

My textbook doesn't contain any solution to the answer so I was wondering if my answer is right. Let v1 =\begin{bmatrix} 1\\ -3 \\ 4 \end{bmatrix} , v2 = ...
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3answers
104 views

Why can't two vectors span $\Bbb R^3$?

I came across a question in my linear algebra textbook and it said: "Given $x_1 = (1, 1, 1)^T$ and $x_2 = (3, -1, 4)^T$: Do $x_1$ and $x_2$ span $\Bbb R^3$? Explain." I'm pretty sure that the answer ...
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7answers
507 views

Does $\{u_1, u_2, u_3, u_4\}$ spanning $\mathbb R^3$ mean that $\{u_1,u_2,u_3\}$ also does? Since it's a subset?

Does $\{u_1, u_2, u_3, u_4\}$ spanning $\mathbb R^3$ mean that $\{u_1,u_2,u_3\}$ also does? Since it's a subset? A little unclear about this...
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272 views

True/False Linear Algebra Vector Spaces/Subspaces

I was doing a bunch of true false for this section and here are a couple I can't seem to understand. The solutions of a matrix equations $Ax=0$ forms a vector space. The set of nonsingular $n\times ...
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2answers
39 views

Which set Spans the Same set

Which set spans the same set as $\left\lbrace(1,2,−1),(0,1,1),(2,5,−1)\right\rbrace$ ? The answers choices each give either a set of 3 vectors or 2 vectors. How will I go about to solve this ...
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1answer
51 views

Vector Spaces Help

Which of the following are vector spaces? $\mathbb Z$, the set of all integers the set of all $2\times 2$ matrices $\mathbb R$, the set of real numbers set of all polynomials $\mathcal C[-1,1]$, the ...
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70 views

Prove that $H$ is a subvector space of $K^{n\times n}$

Let $n\in\mathbb{N}$ with $n\ge 1$ and $K\in\{R,C\}$. $H$ is the set of all $n\times n$ matrices over $K$ whose row and column sums are equal. Prove that $H$ is a subvector space of $K^{n\times n}$ ...
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2answers
73 views

Chart of how the mathematical spaces are related? (soft question)

When dealing with specific function spaces e.g. Sobolev, Hilbert, etc., I find it easy enough to accept the properties of that space and work with them; however, I have a hard time visualizing how ...
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3answers
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Distance from Point on Plane to Origin

I have a practice question that is asking: Find the point on the plane 2x - 3y + z = 3 closest to the origin. What is the distance from that point to the origin? Here is my work so far (if you can ...
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1answer
179 views

Geometrically Describing a Subspace

I have a practice question here and it is asking to geometrically describe this subspace in $R^3$ It is asking if it is a point, a line, or a plane or all of three-space. Here it is: Span {(1,-2, 1) ...