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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Need help understanding wiki's informal description of an affine space.

The following was taken from https://en.wikipedia.org/wiki/Affine_space: The following characterization may be easier to understand than the usual formal definition: an affine space is what is left ...
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On dual spaces and inner products

Let V be a vector space over $\mathbb{C}$ equipped with an inner product $\langle\, , \rangle:V\times V\mapsto\mathbb{C}$. I need to prove that any linear function $\phi:V\mapsto\mathbb{C}$ (element ...
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How to find the basis of the following vector space?

I'm trying to find the basis of the following vector space but I can't seem to be able to find it: $W = \{x = (x_1 , x_2 , x_3 , x_4 , x_5 ) ∈ \Bbb R^5 : x_1 − x_3 − x_4 = 0\}$ I understand that ...
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3answers
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Are the coefficients of a vector according to a basis unique?

If I have a vector space $V$ ( of dimension $n$ ) over real numbers such that $\{v_1,v_2...v_n\}$ is the basis for the space ( not orthogonal ). Then I can write any vector $l$ in this space as ...
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0answers
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isometric quadratic spaces over a prime field

Let $(V, \gamma)$ be a quadratic space, where $V$ is an $n$-dimensional $\mathbb{Z}/(7)$-vector space and $r = r(\gamma)$ is the rank of the bilinear form. I want to show: either, $(V, \gamma)$ is ...
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1answer
23 views

Finding the property of a basis

Let $V = P_2 [x]$, the vector space of polynomials of degree at most 2. Given that $\mathcal B \subset V$, I want to find whether the following is a basis, not linearly independent, not spanning, or ...
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19 views

Each bilinear form induces a unique bilinear form from the dual space

Let $V$ be a finite dimensional vector space over a field $K$. Let $\gamma: V \times V \to K$ be a nondegenerate bilinear form. I now want to show that there exists one and only one bilinear form ...
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finding equality with subspaces direct sum

assume that $U_1 \cap U = \{0\}$ and $U_2 \cap U = \{0\}$ $U_1 \oplus U = U_2 \oplus U$? I thought that it's correct because I could find a counterexample.
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How to find the basis of the following vector spaces?

I'm trying, in vain, to find the basis of the following vector spaces: (a) $W = \{x = (x_1 , x_2 , x_3 ) ∈ \Bbb R^3 : x_1 − 2x_2 + x_3 = 0, 2x_1 − 3x_2 + x_3 = 0\}$ (b) $W = \{x = (x_1 , x_2 ...
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0answers
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Number of two dimensional sub spaces of a vector space over a finite field.

Let {$e_1,e_2,e_3,e_4$} br a basis of $4$-dimensional vector space over a finite field with p elements. The number of $2$-dimensional subspaces of $V$ not containing $e_4$ and not contained in ...
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Help understanding the range and kernel of a linear transformation

I'm having some trouble understanding the Range and Kernel of a linear transformation. The definition goes as follows: Let $T:V \longrightarrow W$ be a linear transformation. Define the sets ...
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Proof of dual norm relation: $\frac{1}{q} + \frac{1}{r} = 1$

Recall: $\|\|$ is a norm in $R^n$, and its dual norm is defined as $\|z\|^*=\text{sup}_{\|x\|\leq1}z^Tx$. If $q$-norm and $r$-norm are dual norm, then we have the following relation: ...
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2answers
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Let $V$ be a $K$ - Vector Space, $f: V \rightarrow V$ a linear map. Under some condition, show that $v, f(v),…,f^n(v)$ is linear independent.

Let $V$ be a $K$ - Vector Space, $f: V \rightarrow V$ a linear map. Let $v \in V$. May a number $n ≥ 0$ exist, so that: $f^n(v) \not= 0$ and $ f^{n+1}(v) = 0$. Show that $v, f(v),...,f^n(v)$ is ...
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1answer
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What is the equation of a facet and it's relation with all points of the polytope?

I was reading about polytopes and I came across about how to define equations of facets defining the polytope. The source I am reading from says If $n$ is a normal vector to a facet $F$ of a ...
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2answers
28 views

Determine interior and boundary of a set

Let $(X,||\cdot||)$ be a normed vector space, where $$X = \big\{ (a_n)_{n \geq 1} ~~|~~ (a_n)_{n \geq 1} \text{ is a bounded real sequence }\big\}$$ and $$\|(a_n)_n\| = \sup_{n \in ...
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2answers
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Matrix ring $M_2(\mathbb{C})$, $\mathbb{C}^2$ with $M_2(\mathbb{C})$-module structure.

Let $R$ be the matrix ring $M_2(\mathbb{C})$. Let $M = \mathbb{C}^2$ with its natural $R$-module structure (just given by the usual action of $2 \times 2$ matrices on $2$-dimensional vectors). My ...
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6answers
78 views

How to prove there's a vector $z \in \mathbb{R}^4$ orthogonal to two linearly independent vectors $x,y \in \mathbb{R}^4$?

Let $x, y \in \mathbb{R}^4$ with $\{x, y\}$ being linearly independent. Prove that there exists a non-zero vector $z$ that is orthogonal to both $x$ and $y$. Any hints on what to do after the ...
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3answers
31 views

Use Vectors To Show Three Vertices Belong to a Right Triangle

The Full Question Theorems Used This is what I call theorem 1: My Work This problem has two major steps as far as I can see. First, I must show that these are points of a triangle(not ...
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1answer
26 views

Understanding the Replacement Theorem (Exchange Theorem)

I'm learning about Basis and Spans and now that's I've figured out what these are, I'm trying to understand the Replacement Theorem(also called the Exchange Theorem). The definition goes like this: ...
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1answer
14 views

Is it possible to find a vector that is orthogonal to this set?

I have a set of four vectors in $\mathbb{R}^4$: $\{ \vec v_1, \vec v_2, \vec v_3, \vec v_4 \}$ The first three are linearly independent, but $ \vec v_4 $ is a linear combination of the others. Is it ...
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2answers
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I'm struggling to find this transformation matrix

$T:\Bbb{P}_3 \to \Bbb{P}_3$ is a linear transformation such that: $$\begin{align} T\left(-2 x^2\right) &= 3 x^2 + 3 x \\ T(0.5 x + 4) &= -2 x^2 - 2 x - 3 \\ T\left(2 x^2 - 1\right) ...
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0answers
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Find a generator for vectorial subspace

S = {$(a, b, c, d) ∈ C^4 : 2ia = b, c + d − ib = 0$} $c+d-i(2ia)=0$ $c+d+2a=0$ $c=-d-2a$ $(a,2ia,-2a-d,d)=a(1,2i,-2,0)+d(0,0,-1,1)$ Is this solution correct?
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0answers
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Cross product of the gradient of two functions

I am having a bit of a confusion with some claims I keep finding on a book of Fluid Dynamics. Let's say we have two functions in 3-D space, $f(\mathbf{x})$ and $g(\mathbf{x})$, with the following ...
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0answers
26 views

Example of the inequality $c_0\neq\bigcup l_p$

As part of an exercise, I was asked to prove or disprove the following proposition: There exists an $x\in c_o$, such that $x\notin l_p$ for every $1\le p\lt\infty$. Before I show my proof, I will ...
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1answer
35 views

Question about conditions for conservative field

Question about conditions for conservative field In common textbooks' discussions about conservative vector field. There is always two assumptions about the region concerned, namely the region is ...
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1answer
30 views

Using SVDs to prove $C(XX^{\prime}) = C(X)$

Let $C$ denote the column space. I would like to prove $C(XX^{\prime}) = C(X)$ for $X \in M_{n \times p}$, $X^{\prime}$ denoting the transpose of $X$. This answer suggests using singular value ...
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1answer
30 views

Find the projection of a vector onto a subspace of $\Bbb R^4$

I need to find the projection of $\vec b = (1,1,1,1)$ onto a subspace of $\Bbb R^4$ described as: $$V=\{(x,y,z,t)\,:\,x=y+t\ \hbox{and}\ 2x=y+z\}\ .$$ Thanks for any help i get guys.
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1answer
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For a matrix $O$ containing columns which are an orthonormal basis for a column space, why does $O^{\prime}O = I$?

Theorem: let $\{o_i\}_{i \in \{1, 2, \dots, r\}}$ be an orthonormal basis for the column space of a matrix $X$ and let $O = \begin{bmatrix}o_1 & o_2 & \cdots & o_r\end{bmatrix}$. Then ...
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0answers
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Why do the vectors perpendicular to [1, 1, 1] and [1, 2, 3] fall on a line, as opposed to a plane?

And what's the intuition here? This is question 6(c) in pset 1.2, Strang's Linear Algebgra, 4th Ed.
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Linear operator of infinite dimension

Let $T: V\rightarrow V$ a linear operator with finite dimension. If exists a linear operator $U: V\rightarrow V$ such that $TU=I$, prove that $T$ is invertible. Prove that if the ...
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2answers
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Prove statement about projection (linear map)

I am working on the following problem and do am not sure how best to approach it. Let $U$ be a vector space over a field $F$ and $p, q: U \to U$ be linear maps. Assume $p + q = \text{id}_U$ and $pq = ...
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3answers
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Understanding the difference between Span and Basis

I've been reading a bit around MSE and I've stumbled upon some similar questions as mine. However, most of them do not have a concrete explanation to what I'm looking for. I understand that the Span ...
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2answers
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Determine which values of $\lambda \in \mathbb{R}$ cause the following vectors to be a basis

I am working on the following problem. Suppose that $\{v_1, v_2\}$ is a basis of a real vector space $V$. For which values of $\lambda \in \mathbb{R}$ is $\{w_+, w_\lambda\}$ a basis of $V$, where ...
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1answer
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What means to complete a pair of vectors $(w, s)$ to an arbitrary basis of $R^d$?

I found in an article this : Let $B = (b_1, b_2, . . . , b_d)$ be an orthonormal basis of $R^d$ such that $<b1, b2 >=< w,x >$ (where $< ... >$ denotes linear span). In order to ...
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1answer
15 views

Difference between orthogonal projection and least squares solution

When you find the least squares solution you solve $$A^TA = A\vec b$$ but to find the orthogonal projection into the "subspace" A, you multiply this result (the least squares solution) with the ...
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1answer
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Kernel Principal Component Analysis (PCA)

I learn kernel PCA from wikipedia. In this article, the eigen equation is \begin{equation} N \lambda \vec{\alpha} = \boldsymbol{K} \vec{\alpha} \end{equation} where $\lambda$ is the eigen value, ...
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1answer
35 views

Complementary subspaces ($K$ and $L$) problem, where $K=ker(p)$ and $L=ker(q)$ with $p,q: U \rightarrow U$ linear maps.

I am struggling with solving the following question: Let $U$ be a vector space over field $F$ and $p,q: U \rightarrow U$ linear maps. Assume $p+q=id_U$ and $pq=0$. Let $K=ker(p)$ and $L=ker(q)$. ...
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1answer
44 views

Determine whether the following set is a vector space

Being pretty new to Linear Algebra, I am trying find whether the set given is a Vector Space or not: \begin{equation*} V = \{A\in M_{3\times3} : AA^{t} = -I\}. \end{equation*} I've tried to solve it ...
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1answer
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Could the Hamel basis of $\mathbb{R^Z}$ be the set $\mathbb{R^Z}-{\mathbf{\{0\}}}$?

This is the follow up question to this question (*) According to page 2 of this link 1 and this link 2, $\mathbb{R^Z}$ (which is referred as $\mathbb{R^\infty}$ in link 1) has elements of the ...
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1answer
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Is $\mathbb{R^Z}$ or its elements countable?

Continue on the self study on infinite vector spaces. According to this link, $\mathbb{R^Z}$ has elements of the following form: $$(y_k)_{k\mathbb{\in Z}}=(\dots y_{-1},y_0,y_{1}\dots)$$ Or more ...
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3answers
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Finding a nullspace of a matrix - what should I do after finding equations?

I am given the following matrix $A$ and I need to find a nullspace of this matrix. $$A = \begin{pmatrix} 2&4&12&-6&7 \\ 0&0&2&-3&-4 \\ ...
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2answers
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linear map $f:V \rightarrow V$, which is injective but not surjective

I am trying to find a linear map $f:V \rightarrow V$, which is injective but not surjective. I always thought that if the dimension of the domain and codomain are equal and the map is injective it ...
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2answers
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Let $C(X)$ be the column space of $X \in M_{n \times p}$. Starting off the proof of $C(X) \cup C(X)^{\perp} = \mathbb{R}^n$

Let $C(X)$ be the column space of $X \in M_{n \times p}$. Prove or disprove the following statement: Every vector in $\mathbb{R}^n$ is in either $C(X)$ or $C(X)^{\perp}$ or both. I interpret ...
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1answer
122 views

What is mathematical structure?

When we have an isomorphism, between 2 groups or vector spaces let us say, then it is said to be structure preserving. An isomorphism exists when there is at least one mutually invertible morphism ...
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19 views

Linear Operator with finite dimension

I'm involved with this exercise. I would greatly appreciate your help Let $V$ be a vector space of dimension $n$ over a field $F$. Let $T: V \rightarrow V$ a linear transformation whose image and ...
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2answers
145 views

How to determine whether a set is a vector space or not?

I'm currently learning Vector Spaces and although I understand the definition of what a vector space is, I can't seem to be able to find the correct answers when doing some questions. I would even say ...
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Quick Vector Space question using the axioms [on hold]

how come this isn't a vector space? (which of the 9 axioms don't hold) W = [ f(1) + f(0) - 5f(-1) = 1 ] Thanks
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1answer
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understanding quaternions - spatial rotations

I would like to know if my understanding about quaternions is correct please: lets say you have a vector in 3d space. You could rotate the x,y and-z frame on a fixed point so that it is parallel with ...
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1answer
34 views

For a given matrix $X$, find two linearly independent vectors in $C(X)^{\perp}$.

Let $$X = \begin{pmatrix} 1 & 1 & 4 \\ 1 & 2 & 1 \\ 1 & 3 & 0 \\ 1 & 4 & 0 \\ 1 & 5 & 1 \\ 1 & 6 & 4 \end{pmatrix}$$ Is there an easy way to ...
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2answers
24 views

Find an orthogonal basis for the space spanned by the columns of the given matrix.

Let $$X = \begin{pmatrix} 1 & 1 & 4 \\ 1 & 2 & 1 \\ 1 & 3 & 0 \\ 1 & 4 & 0 \\ 1 & 5 & 1 \\ 1 & 6 & 4 \end{pmatrix}$$ It is immediately clear to me ...