Questions tagged [vector-spaces]
For questions about vector spaces and their properties. More general questions about linear algebra belong under the [linear-algebra] tag. A vector space is a space which consists of elements called "vectors", which can be added and multiplied by scalars
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Is there a counter example to disprove the following regarding vector addition in binary field?
Let $\{\mathbf{a}_1 , \mathbf{a}_2 , \mathbf{a}_3 , ...., \mathbf{a}_{30}\}\subset \mathbb{F}_2^{15}$ denote the set of binary vectors. Define the set of integers $\{p_k\}_{k=1}^{14}$ as$3 \leq p_1 &...
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The $8$ laws of the definition of vector spaces are not independent. What is the relationship among them? [duplicate]
I am learning linear algebra, and have a question about the eight laws in the definition of a vector space.
A non-empty set $V$ is a vector space on field $F$, if addition $+:V\times V\to V$ and ...
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Questions about how to show $d_1+\cdots +d_n-n+1 \leq {\text{dim}}_k k[x_1,\ldots,x_n]/\mathfrak{a}\leq d_1d_2\cdots d_n\quad $
The following are from Froberg's "Introduction to Grobner bases" , and Hungerford's undergraduate "Abstract Algebra" text.
Background
Theorem 1: $k[x_1,\ldots,x_{n-1},x_n]\...
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What is the connection between bilinear and quadratic forms.
I know that a bilinear form $B$ on the $\mathbb R$-vector space $\mathbb R^n$ is defined to be a map $B:\mathbb R^n\times \mathbb R^n\to \mathbb R$ which is linear in each coordinate.We know that a ...
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Showing that $k[x_1,\ldots,x_n]/\mathfrak{a}$ is a finite dimensional vector space over $k$ assuming basic linear algebra and min amount of abs alg.
The following are from Froberg's Introduction to Grobner bases, and Hungerford's undergraduate Abstract Algebra text.
Background
Theorem 1: $k[x_1,\ldots,x_{n-1},x_n]\backsimeq (k[x_1,\ldots,x_{n-1}])...
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Applying vector decomposition multiple times and RH orthonormal bases
I want to show that I can write any 3D vector $v$ in components with respect to the right handed orthonormal basis $\{e_1, e_2, e_3\}$ (i.e. three perpendicular unit vectors $\{e_1, e_2, e_3\}$ such ...
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Constructing the interval [0, 1) via inverse powers of 2
If $x$ is rational and in the interval ${[0,1)}$, is it always possible to find constants $a_1, a_2, ..., a_n\in\{-1, 0, 1\}$ such that for some integer $n\geq{1}$, $x = a_1\cdot2^{-1} + a_2\cdot{2^{-...
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Is this a valid vector space? (Question Verification) [closed]
Hi - please see the question above. I have a problem with proving associativity i.e. that $(\alpha \beta)v = \alpha(\beta v)$ where $\alpha, \beta \in \mathbb{R}$ and $v \in \mathbb{R}_+$.
I think ...
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Distance between subspaces after premultiplication by diagonal matrice
Let $\mathcal{E}_1$ and $\mathcal{E}_2$ be two k dimensional subspaces in $\mathbb{R}^n$ and two $n\times K$ matrices $E_1$,$E_2$ are basis matrix. Distance between two subspaces is defined to be the ...
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Orthogonal complement with respect to a subspace, and then with respect to the larger space.
Suppose I have the subspaces $W\leq V \leq \mathbb{F}_q^n$, with $n$ finite. Let $\langle ,\rangle\colon\mathbb{F}_q^n\times \mathbb{F}_q^n \rightarrow \mathbb{F}_q$ be the dot-product. If I then take ...
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Bilinear forms and reflexivity [duplicate]
Let $V$ be a finite dimensional vector space over a field $\mathbb{K}$, and let $\varphi: V \times V \to \mathbb{K}$ be a bilinear form on $V$. Let's give some definitions. We say that $\varphi$ is ...
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Tensor Product of a vector space with itself.
I've been reading up on tensor products and have been coming up blank on how to think about $V \otimes_F V$, where $V$ is a vector space over a field $F$.
I only care about what is happening when $\...
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Little trouble understanding a uniqueness proof?
I'm reading Postnikov's Analytic Geometry. Here:
What is happening in there? I understand they may be somehow showing that $x$ is unique but I don't understand what is happening in the equations.
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Does $M_{2,3}(\mathbb{R})$ define a vector space adequately?
I was reading my instructor's notes on vector spaces. I came across the following:
Example 9.2.2 Let $M_{2,3}(\mathbb{R})$ be the set of all $2\times3$ matrices over $\mathbb R$. Show that $M_{2,3}(\...
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Orthonormal basis for $\mathbb{C}^2$ over $\mathbb{R}$ [closed]
$\mathbb{C}^2$ is a 4-dimensional vector space over $\mathbb{R}$ with basis $\left\{\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} i \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}, ...
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How to find basis of vector fields?
I'm figuring out definition of vector fields over a manifold as differentiations of algebra $C^\infty(M)$ of functions on $M$. How can we find their basis starting from this very definition? I know, ...
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Proving the Set of Periodic Functions with Restrictions Form a Vector Space
I understand that a set of periodic functions from $\mathbb{R}$ to $\mathbb{R}$ cannot be a vector space because the set is not closed under the sum of the functions, as discussed here. However, I ...
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Graphical Intuition of a Linear Transformation in terms of Row Vectors
The graphical intuition of a linear transformation (matrix) $A \in \mathbb{R}^{m \times n}$ applied on a vector $\textbf{v}$ in terms of the column vectors $\textbf{c}_i$ of $A$ is quite clear to me:
...
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Stability of Subspaces under a Linear Map in Direct Sum Decomposition
Consider the vector spaces $D_1$, $D_2$, $D$ and $X$ such that $D\subset X$ and $D=D_1\oplus D_2$.
Furthermore, suppose that $L:X\longrightarrow D$ is a linear map such that $D_1$ is stable under $L$...
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Prove that every two lines in space are equal or disjoint or interact at one point only
The question: Let $V$ be a Vactor space over $\mathbb{F}$, Let $\overrightarrow{v},\overrightarrow{w} \in V : \overrightarrow{v} \neq 0 $ . Then we define
$$L_{w,v}=\{\overrightarrow{w}+\...
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Understanding Equivalence of Matrix Elements in Different Bases for Hermitian Operators
Suppose $Q$ and $R$ are two system (which are represented by state vectors in the vector space V) on the same vector space $V$
$|i\rangle$ is an ortonormal base of $V$
$|i_R\rangle$ is an ortonormal ...
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Motivation behind the defination of scalar multiplication of a Vectorspace over a field
In school, we studied physical notations, such as forces, velocities, and accelerations involving both magnitude and direction. We also called any such entity involving magnitude and direction a "...
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Why is the inner product space defined separately?
While learning about the inner product space, I became curious
why it is defined separately?
In my opinion, there seems to be no difference between defining the inner product space separately and ...
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Show that polynomials with a given factor form a subspace
I have a question for one of my assignments but I don't understand how to solve it.
Let $P_n$ be the set of real polynomials of degree at most $n$, show
that
$S=\{p ∈ P_7:x^2+x+4 $ is a factor of $p(...
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Linear Independence without Vandermonde Determinant [closed]
Let $n > 2$ be an integer, $X_1, \ldots,X_n$ be vectors in a vector space, and $\lambda_1, \ldots, \lambda_n$ are nonzero, mutually different scalars.
I want to prove the following implication:
$$
...
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Two definitions of antisymmetrization of a tensor?
I am currently learning about tensors and the exterior product, and I have found some contradictory information. I have seen some sources define the antisymmetrization of a tensor as the following:
...
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Finding Basis for specific Spline Space
Let $S = \{s \in S: s'(a) = s'(b) = 0 \}$ be the spline space that holds all cubic splines with derivate at startpoint (a) and endpoint (b) =0. I want to find a basis for this vector space. I looked ...
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Axler Theorem 5.33: Understanding assumption WLOG
Theorem 5.33 in Axler's book is ($\mathcal{L}(V)$ denotes the set of linear map $V \to V$):
Suppose $\mathbf{F} = \mathbf{R}$ and $V$ is finite-dimensional. Suppose also $T \in \mathcal{L}(V)$ and $b,...
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Can $\text{rank} (T) + \text{nullity} (T) = \dim V$ be proven with this simple argument?
I am helping one of my friends with linear algebra and gave him this theorem to prove as an exercise:
Theorem . Let $V$ and $W$ be vector spaces over the field $F$ and let $T$ be
a linear ...
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Show that $\mathbb{R}[x]_{\leq 2}$ has a two dimensional subspace contained in the orthogonal complement of the subspace
Let $V=\mathbb{R}[x]_{\leq 2}$, and let $f$ the bilinear form given by $f(p,q)=\int_{-1}^{1} xp(x)q(x)dx.$ Find a basis $B$ of $V$ such that $[f]_B$ is diagonal, and show that $V$ has a subspace $U$ ...
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Polynomials as a Linear Combination
I have read that the set of all Polynomials Pⁿ are also a set of vector spaces.
And the explain I read, said that apart from following all the properties of a vector space (identity, communtativity, ...
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Convergence of vector sequence when inner product converges to $0$
Let $d\in \mathbb N_{\ge 2}$, $(v_i)_{i\in \mathbb N}$ be a sequence of vectors in $\mathbb R^d$. If $\inf\limits_{N\in \mathbb N} \sup\limits_{i>j>N} v_i\cdot v_j =0$, does $v_i$ converge to $0$...
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Confusion over tensor definition of exterior power of a vector space and exterior algebra
I am new to and currently learning about Tensor Algebra and Exterior Algebra. I am confused about the definition of the exterior power of a vector space $V$, $\textstyle \bigwedge^k (V)$, and the ...
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Determine which of these four sets are subspaces of the space $X$ and which of these subspaces are closed.
Let $X = C[-1, 1]$ be equipped with the usual maximum norm. Let $Y_1 = \{ f \in X \mid f(-1) = f(1) \}$, $Y_2 = \{ f \in X \mid f(-1) < f(1) - 1 \}$, $Y_3 = \{ f \in X \mid \int_{-1}^1 f(x) dx \...
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Nature of the Euclidean Norm
I've been re-reading my linear algebra book and a definition is given of the norm of a vector in $\mathbb{R}^n$ to be:
||v|| $= \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}$.
For $\mathbb{R}^2$ and $\mathbb{...
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Understanding the implication in linear algebra regarding vectors
Let $V$ be a subspace of $\mathbb{R}^n$ with the usual dot product, and let $\mathbf{z}, \mathbf{w} \in V$ be fixed vectors. If for every $\mathbf{v} \in V$ it holds that $\mathbf{z} \cdot \mathbf{v} =...
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Generalized Rotational Matrix for n-dimensional Euclidean Vector Spaces [duplicate]
$R_{ij}(\theta) := \begin{bmatrix}I_{i-1} & 0 & ... & ... & 0 \\
0 & \cos(\theta) & 0 & -\sin(\theta) & 0 \\
0 & 0 & I_{j-i-1} & 0 & 0 \\
0 & \sin(\...
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Affine Map as a Morphism of Affine Vector Spaces
I've recently took interest in morphism and category theory and I'm amazed how it offers a very general notion. However, I'm struggling to apply this for the affine vector spaces.
I've seen that a ...
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direct sum of general linear space
Let $U_1, U_2, U_3$ be subspaces of a linear space $V$ such that $U_2 \cap U_3 = {0}$ and $U_1 \subseteq U_2$, and $V = U_1 \oplus U_3$. We want to prove that $U_1 = U_2$.
I did the following:
Assume, ...
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Why does the dimension of a subspace equal the dimension of the whole space minus the number of conditions on the set?
let's say we define a subspace $$ W = \{ p(x) \in R_3[x]\,\, | \,\, p(170) = 0\} $$
now it is very hard to determine this subspace's dimension via the condition and to solve all polynomials with ...
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Linear space and vector space correlation
I'm confused with definition of vector space and field.
According to wiki vector_space and field.
Vector space over some field is defined as set of element in $V$ and binary operations that satisfies:
...
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How to interpret $L^2$ norm for functions from $[0,T]\to\mathbb{R}^n$?
I have a function $\alpha \in L^2(0,T;A)$ where $A\subseteq \mathbb{R}^n$.
I understand what it means when $A= \mathbb{R}$, i.e.
$$\Bigg(\int_0^T|\alpha(t)|^2dt\Bigg)^{1/2}<\infty$$
If $A\subseteq \...
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For real matrix $M$ and complex vector $v$, is $M(\operatorname{Re}(v))=\operatorname{Re}(M(v))$? [closed]
Let $M \in \mathcal M_n(\mathbb R)$ be real-valued $n\times n$ matrix, and $v =\operatorname{Re}(v)+ i\operatorname{Im}(v) \in \mathbb C^n$ a complex vector. Does it then hold:
\begin{align}
M(\...
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Is there a finite dimensional vector space over a finite field with exactly two bases?
Is there a finite dimensional vector space over a finite field with exactly two bases?
I searched and found that the answer is NO. But I have an example that $\mathbb{Z}_{3}$ is a 1-dimensional vector ...
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Are the requirements for a field higher than for a vector space? [closed]
My background: finished Linear Algebra 1, learning Linear Algebra 2 as of now.
I find myself appreciating the definition of a vector space and the axioms that it has to satisfy much more, as they make ...
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Prove that the canonical mapping from an infinite dimensional vector space to it's double dual is a one-to-one mapping. [closed]
What is the canonical correspondence from a vector space V to it's double dual $V^{**}.$ Prove that this correspondence is one-one.($V$ need not be finite dimensional)
I tried solving the problem in ...
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How do I find maximal quotients of subspaces of the vector spaces $V_i$?
Say I have linear maps $V_1 \xrightarrow{h} V_2 \xleftarrow{g} V_3 \xrightarrow{f} V_4$. Assume $V_i$ are finite dimensional.
(1) I want to find maximal subspaces $W_i$ of $V_i$ such that in the ...
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What is the dimension of the following vector space? [closed]
I want to find the dimension of quotient
$$
V = \mathbb{R}^{2}{\LARGE /}\left\{\begin{pmatrix} \phantom{-}x\\-x\end{pmatrix} : x \in \mathbb{R}\right\}.
$$
It feels ...
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In an infinite dimensional vector space, is it true that ''a linear transformation is isomorphism if and only if it sends basis to basis.''?
In a finite dimensional vector space, a linear transformation is an isomorphism if and only if it sends basis to basis. I wonder whether this result will also hold for infinite dimensional vector ...
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Spanning set of support functionals in dual space
I am currently studying about supporting hyperplane (or, support functional) in dual space. Since, I am new in these topics I met with the following queries:
Let $X$ be a normed space and $X^*$ be the ...
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