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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conditions of formation of a vector space

Is it compulsory for a linear system to be homogeneous so that the solution space of that system can form a vector space? for example:will the solution space of this linear system form a vector space ...
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Is complex multiplication the only multiplication operation on $\mathbb{R}^2$ that works with the Euclidean norm?

What I'm asking is: viewing complex multiplication as binary operation on $\mathbb{R}^2$, is usual multiplication of complex numbers the only operation $\otimes$ on two vectors $\vec{u}$ and $\vec{v} ...
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20 views

Prove that a specific subset $A$ of a nontrivial vector space $V$ over an infinite field $\mathbb{F}$ is infinite

Let $V$ be a nontrivial vector space over an infinite field $\mathbb{F}$. Suppose $V = \bigcup\limits_{i=1}^{n} S_i$, where $S_i$ is a proper subspace of $V$. We assume that $S_1$ is not included in ...
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1answer
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Defining covectors when the basis is oblique

Given a $2$-dimensional vector space with an oblique unit length basis, say, $(f_1, f_2)$, what is the dual vector or covector corresponding to $f_1$, call it $\hat f_1$? There appear to me to be ...
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1answer
30 views

Angle between planes challenging Question

The plane $r.(a,3,5)=10$ is inclined at an angle of $45^\circ$ to the plane $r.(-5,1,4)$ Find the value(s) of $a$ up to $2$ decimal places. I attempted this problem by forming an equation where ...
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1answer
14 views

Finding a plane which contains a certain line and is perpendicular to another

I have a question on my worksheet which reads the following: 2 lines are given. g1: x= (3,1,3) + t(1,2,-2) and g2: x=(-2,1,-1) + s(0,1,1) the first part is asking if the 2 lines intersect and if they ...
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A special linear transformation

Does there exist infinite dimensional vector space V with a linear transformation on V such that nullityT = rank T = dimV
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Properties of a $3 × 3$ matrix $A$ that contains two equal rows.

A $3 × 3$ matrix $A$ contains two equal rows. State whether each of the following is true or false. (a) $A$ has an inverse. (b) The rows of $A$ are linearly independent vectors. (c) The determinant ...
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Find linearly independent vectors formally

How can I find $3$ vectors $a$, $b$ and $c$ in $\mathbb R^3$ such that $\{a, b\}$, $\{a, c\}$ and $\{b, c\}$ are each linearly independent sets of vectors, but the set $\{a, b, c\}$ is linearly ...
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1answer
22 views

Finding orthonormal basis of a polynomial (without a given dimension)

I have the following problem: Find an orthonormal basis of the plane $x_1 + 6x_2 - x_3 = 0$. I am suspecting that this problem is asking me to apply Gram-Schmidt so I can get $q'_1$ and $q'_2$. ...
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1answer
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Intersection of linear Transformation.

Good night, i was thinking about this: If $T(L_{1}+L_{2})=T(L_{1})+T(L_{2})$, then i can work with this: $T(L_{1}\cap L_{2})=T(L_{1})\cap T(L_{2})$ but I can not think of anything to prove this ...
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Which one is the correct definition of natural norm?

In the definition 2 of Normal Subgroup Reconstruction and Quantum Computation Using Group Representations, the authors have defined the natural norm of a matrix as follows. The natural norm of the ...
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1answer
32 views

Is the space $\mathbb{R}_+\times S\times S$ linear?

The space $\mathbb{C}$ (or even $\mathbb{R}^2$), which is a linear space over $\mathbb{R}$, can be obtained from the Cartesian product $\mathbb{R}_+\times S$ by gluing to the point the layer $0 ...
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57 views

show that a vector is on a line

can you please explain this question to me? Show that for any two vectors (vector a) and (vector b), the (vector a + vector b)/2 is on the line that connects the vector a and the vector b. show that ...
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1answer
18 views

Distance between two lines

Find the distance between the lines $l_1:$ $x=1+4t,y=5-4t,z=-1+5t$ and $l_2:x=2+8t,y=4-3t,z=5+t$ So the approach in general is to find a vector that is orthogonal to 2 planes that the lines are ...
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3answers
37 views

Distance between a plane and a point

I understood that for finding a distance between a plane and a point we first find a vector between a point on a plane and the given point and then take the projection on the normal vector. Is ...
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2answers
34 views

How will the basis vectors of the subspace $\mathbb{R}^n$ consisting of those vectors $A=(a_1,\cdots,a_n)$ such that $a_1+\cdots+a_n=0$ look like?

How will the basis vectors of the subspace $\mathbb{R}^n$ consisting of those vectors $A=(a_1,\cdots,a_n)$ such that $a_1+\cdots+a_n=0$ look like? The initial problem was "what is the dimension of ...
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1answer
32 views

Compactness in a vector space

If $E$ is a normed space and $F$ is a subspace of $E$, how to prove that if $F\neq\{0\}$ then $F$ is not compact? I begin by this let $x\in F$ then $F=\bigcup_{x\in F} B(x,\varepsilon)$ how to say ...
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1answer
56 views

Determine if the Set of Vector is a Subspace of $\mathbb{R}^n$

Can you help me check whether what I did is right or wrong? Here are the questions: Which of the following sets are subspaces of $\mathbb{R}^n$? (a) The plane of $\mathbb{R}^3$ that passes ...
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Representation decomposition over $GL_2(\mathbb{C})$

I have found that $Sym^2(V) \otimes Sym^2(V)$ can be decomposed over the special linear group as follows: $Sym^2(V) \otimes Sym^2(V) \simeq Sym^4(V) \oplus Sym^2(V) \oplus 1$ This is done using the ...
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1answer
19 views

Understanding the dimension of a particular subspace in Linear Algebra in C³

I am having trouble to determine the dimension of the subspace of T. How is it done, when there is just one vector given?
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1answer
45 views

Getting an isomorphism from a short exact sequence of inner product spaces

Let $L,M,N$ be finite dimensional inner product spaces and $0 \to L \xrightarrow{\alpha} M \xrightarrow{\beta} N \to 0$ is a short exact sequence. Now let $\beta^* : N \to M $ be the adjoint map (the ...
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Free graded k[x] modules have homogeneous bases

I was reading the article "Cary Webb. Decomposition of graded modules. Proceedings of the American Math- ematical Society, 94(4):565–571, 1985" where in the beginning "Free graded k[x] modules have ...
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3answers
40 views

What is the space $\operatorname{Sym}^2(V)$ and how does it act on the vector space $V$?

If $V$ is a vector space over $\mathbb{C}$ with basis vectors $e_i$, what is the space $\operatorname{Sym}^2(V)$? I am hoping someone can give me some insight into this space; perhaps by ...
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1answer
28 views

Can you construct a basis for an infinite dimensional vector space from a set of vectors that span that space?

Suppose I have an infinite dimension vector space V (not necessarily countably infinite). Suppose a have a set S that spans the space. If V is finite dimensional, it is trivial to construct a basis ...
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2answers
40 views

geometic description of vector span

i am having some poblem with this question Show that the set if vector is linearly dependant. If the vector span a line, provide the equation of the line. If the vector span a plane, provide the ...
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0answers
25 views

Suppose we have three vectors $x,y,z\in \mathbb{R^3}$

Suppose we have three vectors $x,y,z\in \mathbb{R^3}$ Further suppose that x ⊥ y and y ⊥ z then is x ⊥ z?
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2answers
22 views

What does this symbol means in Banach's spaces?

Let $X,Y$ be non trivial vector spaces such that $L(X,Y)$ is Banach's $\Rightarrow Y$ is Banach's. I'm missing the definition of $L(X,Y)$ in my notes, also can you give me a hint? I'm new in Banach's ...
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Prove that this transformation inverse exists and it's bounded

If $X,Y$ are Normed Vectorial Spaces, $T$ is a bounded lineal transformation. Prove that if exists $b>0$ such that $\|Tx\|\geq b\|x\| \forall x\in X$. Then $T^{-1}:Y\rightarrow X$ exists and it's ...
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How to prove that 1, sin(x), xsin(x) and cos(x) are linearly independent?

My attempt: Suppose that $a\cdot 1 + b\cdot \sin(x) + c \cdot x\sin(x) + d\cdot \cos(x) = 0$. Let $r = \sqrt{b^2 + d^2}$ and $\tan \phi = \frac{b}{d}$ Then $a + r \cdot \sin(x + \phi) + c\cdot ...
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3answers
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If $\dim(V) $ is infinite, show that $V\oplus V$ is isomorphic to $V$

For a vector space $V$ of infinite dimension, to show that $V\oplus V$ is isomorphic to $V$ is to show that there exists an invertible linear transformation between $V \oplus V $ and $V$. Every ...
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Bivariate normal distribution (check)

I need to determine probability. Random variable X has a bivariate normal distribution with mean vector μ and covariance matrix Σ. $$X = {x_1 \choose x_2}, \mu = {-2 \choose 7}, \Sigma = ...
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Minimizing matrix norm via left-multiplication by $SL(m)$

Suppose that $M$ is an $m\times n$ matrix of full row rank, with $m \leq n$. Then if $\|M\|$ is the matrix norm induced on $M$ from the norm on our vector space, we can look for the following ...
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Find the all generalized eigenvectors of T

Let Define $T\in L(\mathbb C^2)\;by\\ T(w,z)=(z,0).$ Find the all generalized eigenvectors of T the standard basis of $C^2$ is {(1,0),(0,1),(i,0)(0,i)} T(1,0)=(1,0) then how can we write ...
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Find a volume, by parametrization sphere..

Consider a parametrization $C$ in $\mathbb R^3$ space, $x=\frac{4}{5}\cos t, y=1-\sin t, z=-\frac{3}{5}\cos t\;$ for some $t$. For the point $P$, moving on $C$, and the fixed point $Q$, moving on ...
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0answers
34 views

Color space as a vector space

I am not sure that this is the best place for this topic, so I apologize in advance. I have two questions. I think that color space with say additive colors (red, green, blue) forms a vector space. ...
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References for Duality Theory

I was wondering if anyone had any recommendations for Duality Theory. I've touched on Duality before in various courses but it's coming up quite a lot in my studies at the moment. I guess what I'm ...
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36 views

If $W$ is a subspace of $V$,then $\dim(W)+\dim(W^0)=\dim(V)$.

From the structure of this all i getting is that If $V$ an n-dimensional vector space with an ordered basis $\beta=(x_1,x_2,x_3,\dots,x_n)$ among them (say) first $k$-vectors form the basis ...
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1answer
37 views

Taking curl of Euler equation

Consider an inviscid incompressible flow. Euler’s equation can be written as $$\frac{\partial \textbf u}{\partial t} + \textbf ω × \textbf u = −\textbf∇\bigg( \frac pρ + \frac 12 \textbf u^2 + V ...
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1answer
30 views

How to prove if there exist unique $u$ and $w$ such that for any $v$, $v=u+w$, then $V$ is the direct sum of $U$ and $W$

How do I prove the statement: if there exist unique $u$ and $w$ such that for any $v$, $v=u+w$, then $V$ is the direct sum of $U$ and $W$? ($U,W,V$ are vector spaces, $u \in U, w \in W, v \in V$) I ...
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1answer
45 views

Basis and dimensions example

Every basis of $\mathbb R^6$ can not be reduced to a basis of $5$-dimensional subspace of $\mathbb R^6$ by removing one vector . Can anyone give an example for that?
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1answer
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Show that $\int r\cdot n ds$ equals three time the volume of $\omega$.

Let $\Omega$ be an open region in $\mathbb{R}^3$ with surface $∂\Omega$ on every point $P$ of which the unit outward pointing normal $n = n(P)$ is well defined and smoothly varying. Let $r = (x, y, ...
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1answer
41 views

What do you call this equivalence relation? $A \simeq B$ if $A = P^t BP$ for some invertible matrix $P$

If $A, B$ are square matrices with coefficients in some ring, we say that $A$ is similar to $B$ if $A = PBP^{-1}$ for some invertible matrix $P$. Similar matrices represent the same linear operator ...
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1answer
37 views

Given two lines in Cartesian form, find the vector equation of a line which passes through the intersection of two lines.

Given two lines in Cartesian form, find the vector equation of a line which passes through the intersection of two lines (and is perpendicular to both). No points given just the two equations. What ...
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1answer
81 views

Null space of matrix A and column space of transpose matrix A

Let A be an m×n matrix. Show that every vector v $\in R\ {^n} $can be written uniquely as w + u, where w is in the null space of A and u is in the column space of transpose A
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Symplectic form on $\mathbb R ^{2n}$

What are all symplectic form $\omega$ on $\mathbb R^{2n}$. Where, a ''symplectic bilinear form'' on $\mathbb R^{2n}$ is . a bilinear form: a map $\omega: \mathbb R^{2n}\times \mathbb R^{2n}\to ...
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1answer
22 views

Difference between $\mathbb{R}^4$ and $\mathbb{C}^4$ in subspace spanned by some vectors.

This is a problem in Hoffman / Kunze, Linear Algebra: Let $$\alpha_1=(1,1,-2,1), \quad \alpha_2=(3,0,4,-1), \quad \alpha_3=(-1,2,5,2).$$ Let $$\alpha=(4,-5,9,-7), \quad \beta=(3,1,-4,4), \quad ...
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1answer
32 views

Dimension and sum of linear subspaces

How to find the dimension of the intersection and the sum of linear subspace defined as linear span of the vectors systems. Vectors: $$ \begin{align} V_1&=\langle (1,1,0,1,1), ...
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1answer
82 views

Isomorphism between endomorphism algebras

Assume that $R$ and $S$ are associative $\mathbb{C}$-algebras with unit $1_R$ and $1_S$, respectively. In addition, assume that $_RM$ is a simple left $R$-module and $_SN$ is a simple left $S$-module. ...
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1answer
30 views

Prove the image of basis elements is linearly independent

I was wondering if someone could give me a quick proof or counterexample to the following statement. Let $f:V \rightarrow W$ be a linear map between finite dimensional vector spaces $V$ and $W$, both ...