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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Dimension of a vector space $U = \{(x,y,z)\in\mathbb{R}^{3}: x-y+2z=0\}$

Let $U = \{(x,y,z)\in\mathbb{R}^{3}: x-y+2z=0\}$ My lecturer says without explanation: 'Intuitively $U$ has dimension $2$'. Can someone explain his intuition?
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Prove that $\lim_{t\to t_0}[f(t) \times g(t)]=u \times v$

Let $f(t)=(f_1(t),f_2(t),f_3(t))$, $g(t)=(g_1(t),g_2(t),g_3(t)).$ $$\lim_{t\to t_0}f(t)=u; \lim_{t\to t_0}g(t)=v.$$ Prove: $$\lim_{t\to t_0}[f(t) \times g(t)]=u \times v$$. Thanks ahead:)
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32 views

Verifying The Divergence Theorem

Q: "Given the cylindrical region $x^2 + y^2\leq 1 $ ,where $ 0\leq z \leq 1 $, and the vector field $\underline{F} = 3x\underline{i} -5z\underline{k} $, verify the divergence theorem." For the ...
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3answers
46 views

Vectors and orthonormal basis vectors help!

I'm not entirely sure how to go about answering this question about vectors. Any advice/help is appreciated. Write the vector $\displaystyle a =\begin{bmatrix}3\\-1\\7\end{bmatrix}$ as a linear ...
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1answer
35 views

Convergence of vector spaces

I'm considering vector spaces over $\mathbb{R}$ or $\mathbb{C}$ : $(E_n)_n$ is a sequence of vector space (all included in $\mathbb{R}^p$ for example, with $p$ fixed). What meaning(s) do we usually ...
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1answer
240 views

Show that S (a subset of V) is contained in span(S)

Let $\text{span}(S) = \lbrace v \in V \mid v\ \text{is a linear combination of vectors in}\ S\rbrace$. I need to show that $S$ is contained within $\text{span}(S)$. I know if $S$ is nonempty, $0$ is ...
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1answer
67 views

Definition for union of spans

Let $K, L$ two sub-sets of $V$, a vector space. Consider: $$Sp(K)\cup Sp(L)$$ What is the right definition for the union of those spans? Is it: $$\{v\in V : v=\alpha\cdot k + \beta\cdot l \}$$ ...
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1answer
265 views

number of planes possible such that it is equidistant from 4 non coplanar points

If there are for non coplanar points find the number of planes such that all four of them are equidistant from the plane . Sorry one of those problems where dont know what to do . How should i do this ...
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31 views

Find the equation of the plane

This is my first post to math.stackexchange - hope you guys can help me. I'm preparing for a linear algebra exam this week, and would like to check that my methods are correct. The question asked in ...
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1answer
17 views

Defining a line of two equations in three varibles for subspaces.

$U=\{(x,y,z)\in\mathbb R^3:x−y+2z=0\}$ is a vector space. Define the line $V=\{(x,y,z)\in\mathbb R^3:−x=y=z\}$. The set $V$ is seen to be a line as the points in $V$ satisfy two linear equations in ...
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3answers
121 views

What's the term for a “physical vector space”?

In physics, we often use the term "vector space" (or just "space," or other similar terms) to refer to a vector space in which the different dimensions are "compatible," i.e. that it "makes sense" to ...
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1answer
624 views

Finding a transition matrix $P_{B'\leftarrow B}$ and using it to find $B'$ coordinate matrix

Here are my full instructions: Find the transition matrix $P_{B'\leftarrow B}$ from $B=\left\{\left(1,1,\right),\left(2,3\right)\right\}$ to $B'=\left\{\left(1,2\right),\left(0,1\right)\right\}$, ...
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612 views

Find the resulting speed and direction. Trig Problem involving resultant and vectors.

A barge is pulled by two tugboats. The first tugboat is traveling at a speed of 15 knots with heading 130°, and the second tugboat is traveling at a speed of 11 knots with heading 190°. Find the ...
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1answer
539 views

Problem involving Bearing, Heading, and True Course.

A plane is flying with an airspeed of 170 miles per hour and heading 150°. The wind currents are running at 30 miles per hour at 170° clockwise from due north. Use vectors to find the true course and ...
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4answers
62 views

Does the following set span $R^3$ and is it a basis of $R^3$?

$\left\{ \left( 1,1,1 \right), \left( 0,1,1 \right), \left( 0,0,1 \right), \left( 1,2,3 \right) \right\}$ The first thing I did was test for linear independence: $x=(0,0,0)$ $u=(1,1,1)$ $v=(0,1,1)$ ...
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1answer
101 views

Testing invertible matrices for closure under addition

Determine whether the following subsets of V are subspaces of V. Justify your answers: if the subset is a subspace, verify the necessary properties a subspace must have; if it is not a subspace, ...
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1answer
61 views

Testing a subspace for closure under addition

Determine whether the following subsets of $V$ are subspaces of $V$. Justify your answers: if the subset is a subspace, verify the necessary properties a subspace must have; if it is not a ...
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1answer
66 views

If $V/W\cong C$ is it true that $V=W\oplus C$ ?

We know that for a vector space $V$ and its subspace $W$ if $V=W\oplus C$ then the quotient space $V/W$ is isomorphic to a subspace of $V$ (namely, $C$). Is the inverse true?
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total order on finite dimensional vector space over $\mathbb{R}$

We know that, If we start with a basis of a finite dimensional vector space $V$ over $\mathbb{R}$ by using lexicographic ordering with respect to that basis, we have a total ordering $\lt$ on $V$ ...
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43 views

The union of line $\bf{L}$ and plane $\bf{P}$ is also a subspace of $\mathbb{R^3}$, right?

Assume a vector space $\mathbb{R^3}$, in which a plane $\bf{P}$ and a line $\bf{L}$ which are passing through the origin vector $\bf{Z^3}$. It can be easily seen that $\bf{P}$ and $\bf{L}$ are ...
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1answer
85 views

Tensor Product of Vector Spaces - Quotient Definition

I'm trying to figure out exactly what the tensor product of vector spaces is. This is what I understand so far: If $V, W$ are vector spaces over a field $R$ then the free vector space $C(V\times W)$ ...
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0answers
48 views

can I show this set is closed? [duplicate]

Suppose $X$ is a normed vector space with norm $||\cdot||$. Suppose $S$ is a linear subspace of $X$ and $S$ is closed. Let $x\in X$. I was wondering whether the set $\{||x+s||: s\in S\}$ is closed in ...
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1answer
85 views

${n \brack k} = \sum_{m=0}^{k}2^{(k-m)^2}{n-k \brack k-m}{k \brack m}$

$${n \brack k} = \sum_{m=0}^{k}2^{(k-m)^2}{n-k \brack k-m}{k \brack m}$$ I need hint to prove this. ${n \brack k}$ is the number of $k$ dimension subspaces of $n$ dimension space over field $F_2$. I ...
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1answer
31 views

How to find the vector of length $5$?

Given vector $\mathbf a = -2\hat i + \hat j + \hat k$ and $\mathbf b = \hat i - 3\hat j + \hat k.$ Find the vector of length $5$ perpendicular to the plane containing vector $\mathbf a$ and $\mathbf ...
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0answers
54 views

How to complete one vector to create a basis of a certain subspace?

I have a subspace of $R^3$ that its basis is the vector $(0,1,1)$. I want to add vectors to it, such that they will create a basis of a subspace, that is equal to the subspace that its basis is ...
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0answers
36 views

Getting X and Y component vectors

This diagram shows that to get a component vector, you use the "SOH CAH TOA" rule in a right angle triangle, Additionally, this is how I learned to do it many years ago. However, in my math ...
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2answers
67 views

Extending the Intersection of Subspace

For two subspace, one can express the dimension of the sum as $$ \dim(U_1 + U_2) = \dim U_1 + \dim U_2 - \dim (U_1 \cap U_2).$$ However, the obvious extension to three subspacess fails, in the ...
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1answer
61 views

Integral Curves of a Vector Field

How do I find the integral curves of a vector field and what are they intuitively? eg. what are the integral curves of vector field $X=\frac{1}{x}\frac{\partial}{\partial ...
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3answers
38 views

Some questions of vectors and dense subsets

I have a couple of quick functional analysis related questions: 1.Say we have a normed space $V$ and reflexive, separable Banach space and $K \subset V$ a closed, convex, bounded subset of $V$. ...
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2answers
170 views

Does an orthogonal transformation always have an orthogonal matrix?

Suppose we have a linear operator $T:\Bbb F^n -> \Bbb F^n$. If we fix the basis as the standard basis for both the domain and the co-domain then it turns out that the matrix of an orthogonal ...
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0answers
32 views

Jacobian in Change of Variables

Let us consider an integral $\int \mathrm{d} ^ 4 k _ {2} \int \mathrm{d} ^ 4 k _ {1} \, f (k _ {1}, k _ {2})$, where $k _ {1}$ and $k _ {2}$ are four-dimensional vectors in Euclidean space. We want to ...
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2answers
57 views

$(v_1,v_2\ldots,v_n)$ is linearly independent. Can $(v_1+u_1,v_2+u_2,\ldots,v_n+u_n)$ remain linearly independent?

Given are $n$-dimensional vector space $\langle R^n;+\mid R\rangle$ and $n$ vectors $(v_1,v_2,\ldots,v_n)$ which are linearly independent, $u_i\in R^n, i=\overline{1,n}.$ Say, $x$ = ...
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2answers
99 views

Writing a vector as the sum of two other vectors.

Suppose you have 2 vectors $\vec a = (1,1,2)$ and $\vec b = (3,4,-2)$, how would you write $\vec a$ as the sum of 2 vectors $\vec c$ and $\vec d$ where $\vec c$ is in the direction of $\vec b$ and ...
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1answer
163 views

How do you pronounce $\mathbb{R}^2, \dots, \mathbb{R}^n$ [closed]

I've been starting some vector calculus and I keep pronouncing it "R squared" in my head, which is clearly wrong. What is a better way to pronounce it, so that I can state things like $f: \mathbb{R} ...
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1answer
69 views

Linear Algebra,Conjugate Transpose

Let $ M_n(\mathbb C) $ be the space of all $ n\times n $ matrices with complex entries. Prove that function $ \langle, \rangle : M_n(\mathbb C) \times M_n(\mathbb C) \to \mathbb C $ defined by $ ...
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3answers
41 views

Linear operator from F^n to F^n without reference to any basis

Let $T:F^n \to F^m$ be a linear operator defined by $T(v)=Av$ where $A$ is any $m\times n$ matrix and $v$ is an element of $F^n$. Does this transformation make any reference to some basis of $F^n$? I ...
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3answers
22 views

Show the following is a subspace and find its dimension

If $V=K^{2009}$ where $K$ is a field. Show $W=\{(a,b,a,b,a,b,...)|a,b \in K \}$ is a subspace and find $dim_{K}W$. My Attempt; For $W$ to be a subspace of $V$ two propeties must hold; Closure by ...
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1answer
80 views

linear space $\{MN-NM|M,N\in M_n(\Bbb C)\}$

Let $M_n(\Bbb C)$ be the linear space of all $n\times n$ complex matrices, then 1). the set $\{MN-NM|M,N\in M_n(\Bbb C)\}$ is a subspace of $M_n(\Bbb C)$; 2). $\{MN-NM|M,N\in M_n(\Bbb C)\}=\{A\in ...
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1answer
99 views

$\mathrm{col}(AB) = \mathrm{col}(B)$

Let $A$ be a real invertible $n\times n$ matrix, and $B$ a real $n\times m$ matrix. $\mathrm{col}(.)$ denotes the column space of a matrix. What are the conditions for $\mathrm{col}(AB) = ...
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0answers
31 views

Show that a function $\phi:(V, \|\cdot\|_{C^1}) \rightarrow (W, \|\cdot\|_{\infty})$, $f \rightarrow f'$ is continuous

I have the following: Show that the function $$\phi:(V, \|\cdot\|_{C^1}) \rightarrow (W, \|\cdot\|_{\infty}),~~f \rightarrow f'$$ is continuous. With $$\|\cdot\|_\infty = \sup\{|f(x)| ~ \big| ~x\in ...
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0answers
43 views

Classify all possible $R$-module structures on a vector space

Let $V$ be an $n$-dimensional complex vector space. In particular, it is an Abelian group. Let $R$ be a (commutative, unitary) $\mathbb C$-algebra. Problem. I would like to parameterize all ...
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115 views

What does the vector space $\mathbb{R}^{\mathbb{R}}$ look like?

I can imagine $\mathbb{R}^{\mathbb{N}}$. For instance, the set of real series is part of this space, as is any infinite (but discrete numbered) tuple of reals. But how can I imagine ...
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1answer
114 views

Show that a square with vertices t, u, v, w has center 1/4 (t+u+v+w).

I need a help with this question! Show that a square with vertices t, u, v, w has center 1/4 (t+u+v+w).
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1answer
36 views

Help find the equation of two planes

I have the question Consider the line L through the distinct points A = (a,b,c) and D = (d,e,f) Find the equations of the two planes which intersect at right angles along L MY ATTEMPTED SOLUTION I ...
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1answer
29 views

How do I prove the following equality properly?

In a linear algebra assessment, I had to show that $W$ is subspace of $\mathbb{R}^3$ for: $$W = \left \{ (x,y,z) \in \mathbb{R}^{3} \mid \frac{x}{3} = y = 2z \right \}$$ I showed that it is a ...
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0answers
25 views

Proof of equivalance beginning with the geometrical definition of the scalar (dot) product and the algebraic definition of the dot product.

Geometrically the dot product is defined to be |x| |y| cosθ where |x| |y| are the magnitudes of the two vectors in question. Can anyone prove, beginning with this geometrical definition, an ...
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0answers
42 views

Field of vector fields

For every point $A$ outside a sphere with radius $a$, there's a field $$F= \frac{K}{r^4d^2} $$ where $r$ is distance between point $A$ and the center of the sphere, and $d$ is distance between point ...
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79 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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2answers
26 views

If a lot of vectors from vector space is eliminated, is it possible for remaining nonempty set to remain a vector space?

Well, I think it can because that way we would get some sort of vector subspace and by definition vector subspace is indeed a vector space. Even if we would eliminate all the vectors we would still ...
3
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3answers
144 views

Prove that $\mathbb{Z}$ is not isomorphic to additive group of any vector space over any field.

Prove that $\mathbb{Z}$ is not isomorphic to additive group of any vector space over any field. Proof. Surpose that: $\phi : (A, +) \rightarrow \mathbb{Z} $ is an isomorphism. Then there is some ...