# Tagged Questions

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### On the canonical isomorphism between $V$ and $V^{**}$

I am trying to understand more about the Bidualspace (or double dual space). The whole idea is that $V$ and $V^{**}$ are canonically isomorphic to one another, which means that they are isomorphic ...
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### Linear maps and matrix coefficients

I am currently working through this page in my script: Can somebody explain what this means and how it works in practice? Perhaps if I saw an example I could follow it. Thanks for your help!
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### Notation - Transpose of Block Matrices [Lay P121 Q2.4.12]

Definition of Transpose is $(A^T)_{ij} = A_{ji}$ $1.$ Why $\begin{bmatrix} M & N \end{bmatrix}^T = \begin{bmatrix} M^T \\ N^T \end{bmatrix}$, and NOT $\begin{bmatrix} M \\ N\end{bmatrix}$? ...
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### Orthogonal Projections: Definition + Characterization

Hey there I'm hanging at so many little steps when proving: $P\text{ orthogonal projection}\iff X\text{ orthogonal decomposable}$ My problem is there is simply missing a precise definition of what an ...
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### Orthonormal Bases

I am struggling to get my head around orthonormal bases, this is the defintion in my course notes: If anyone could clarify/explain the concept to me, it would be much appreciated. I am a university ...
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### Rank of a Matrix under certain conditions

I am a little confused about the rank of a matrix. When does the rank of a matrix equals to zero? Is rank of a matrix equal to zero when it is a zero matrix or the matrix has no elements in it? Thank ...
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### $\hom(V,W)$ is canonic isomorph to $\hom(W^*, V^*)$

Introduction My Semester just started and we have a new Professor for Linear Algebra II (replacing our former Professor). Apparently we are behind our schedule and thus we only had a brief ...
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### Definition of “point” and “vector” in $\Bbb{R}^n$, and a model for $\Bbb{A}^n(\Bbb{R})$..

Can I use the following definitions? let $(a_1,a_2,..,a_{n+1}) \in \Bbb{R}^{n+1}$, $(a_1,a_2,..,a_{n+1})$ is a point of $\Bbb{R}^n$ if $$a_1=1 \wedge \forall i \in \{2,...,n+1\} (a_i \in \Bbb{R})$$ ...
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### About definition of “direct sum of $p$-vector subspaces”

In the books 1 and 2, in Somme directe d'une famille de sous-espaces vectoriels, I am reading the following: 1) let $E,F$ two vector subspaces of $V$, $E+F$ is direct sum, $E+F \doteq E\oplus F$, if ...
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let $f \in End_K(E)$, and $g: (E \times E)\to \Bbb{R}^1$ a symmetric bilinear form positive definite, $f$ is self-adjoint endomorphism if $$\forall v,w \in E(g(f(v),w)=g(v,f(w)))$$ It is correct? ...
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### Definition of “symmetric bilinear (real) form indefinite”

In my studies I use these definition: Def.: $f \in \mathscr{B}_ \Bbb{R}((e \times e), \Bbb{R})$, $f$ is symmetric bilinear (real) form positive definite if 1) $\forall x \in e(f(x,x)\geq0)$ 2) ...
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### Equivalent definitions of isometry

Consider a map $T:\mathbb{R}^2\to\mathbb{R}^2$ such that $\lVert T(x)\rVert=\lVert x\rVert$. Is this equivalent to stating that $\langle x, y\rangle=\langle T(x), T(y)\rangle$ for all ...
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### Difficulty with terminology/standard definition for Jordan-normal form.

(Perhaps this is something rather for MOverflow, don't know) I've understood the concept of the Jordan-normal-form such that it is similar to the idea of diagonalization of a matrix, but can be ...
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### Is the empty matrix $M$ ($m \times n$) with $m$ or $n$ equal to zero in RREF ? Does it posses every matrix property one would wish?

Is the empty matrix $M$ ($m \times n$) with $m$ or $n$ equal to zero in RREF ? Reading a proof in Linear Algebra induction starts off by proving $P(n)$ holds for $n=0$, where $P(n)$ is the ...
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### What is the difference between a point and a vector

I understand that a vector has direction and magnitude whereas a point doesn't. However, the course note that I am using states that a point is the same as a vector. Also, can you do cross product ...
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### Transformation-Matrix (Definition & explanation)

I have to do a proof ("Let V be a vector space with basis A, B, C. Show that $T_{AC} = T_{BC}T_{AB}.$ Well, I really don't know what is menat by $"T_{AB}"$ or something like this. I thought about the ...
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### confusion about rank and nullity of matrix and rank-nullity theorem

I can see that rank + nullity = number of columns of the matrix. Does that mean the dimension of matrix is the number of columns? Isn't dimension of a $m\times n$ matrix $mn$? Thanks.
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### Subspace with different vector space operations

Let $A,B$ be vector spaces such that $A\subseteq B$. Is it true that $A$ is a subspace of $B$? I claim that the answer is no, because it is possible that $A$ and $B$ might be equipped with different ...
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### Definition of determinant [closed]

When determinant is a funtion from the set of all nxn matrices to a scalar, how can I define the definition of determinant? What characterizes the determinant function?
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### Adjoint of a Matrix Definition

Tom M. Apostol in his book "calculus Vol. 2" page 122 (see image below) defines adjoint of a matrix as the transpose of the conjugate of the matrix. Is this definition always correct ? Does it agree ...
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### Is the definition of linearity redundant?

For some two functions f(x) and g(y) and for the transformation T, T is linear if: ...
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### Quotient spaces in linear algebra

There's a statement in some notes I'm reading that goes like this: "...$V/U$ is a 'simplified version' of $V$ where the elements of $U$ are ignored" ($V$ and $U$ are vector spaces). I'm still ...
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### Similar linear operators and change of coordinates

Let $S, T$ be operators in $\mathcal{L}(V)$, the space of all linear maps from $V$ to itself. In my lecture notes, I have the definition of similar: "We say that operators $S,T \in \mathcal{L}(V)$ ...
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### What does it mean when a set is said to be a “finitely generated vector space”?

I've somehow managed to go through 3 years of my Maths degree without truly understanding what the term "finitely generated vector space" means. Generated by what? Generating what? And for what? I ...
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### About definition of endomorphism on vectos space

"let $f$ be a homomorphism between two vector spaces $V$ and $W$, $f$ is endomorphism on $V$ if $im(f) \subseteq V$" is correct? Thanks in advance!
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### How to Define an Infinite Anti-Diagonal Matrix

We can define an infinite diagonal matrix with some ease, and then say that a finite diagonal matrix is the top left sub-matrix of our infinite one. Can we define an infinite anti-diagonal matrix? It ...
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### Apostol question on alternative definition of dot product

The problem says: Suppose we define the dot product by $A\cdot B = \sum_{k=1}^n |a_kb_k|$. Which of the following properties hold with this new defition? Does the Cauchy-Schwarz inequality still ...
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### Perpendicular Symbol as Matrix Superscript

If $A$ is a matrix that is not (necessarily) square, then what is $A^\perp$? What I do know is: $A^\perp$ is a matrix, not the orthogonal complement It is related to the QR Decomposition. And ...
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### What does it mean for the determnant of a matrix to be independent of the vector space

Explain what it means for the determinant of the matrix, representing an operator $F$, to be independent of the basis of the vector space. Prove this property of the determinant. I'm not exactly ...
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### Property of Division by vector for a field

Serge Lang in "Linear Algebra" on page 2 says that The essential thing about a field is that it is a set of elements which can be added and multiplied, in such a way that additon and ...
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### Meaning of $L_A$?

Let $A$ be an m*n matrix with entries from a field $F$. $L_A: F^n \rightarrow F^m$ defined by $L_A=Ax$. I'm a bit confused about this definition. $L_A$ is a matrix representation of linear ...
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### Difference between Spanning set and Postitive Spanning Set

I do understand the difference as mentioned in the texts about spanning set and positive spanning set, im somehow missing how if $v_1.. v_r$ is a positive spanning set for $R^n$, then $v_2 ... v_r$ ...
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### What is a bijective linear mapping called?

Friedberg - Linear Algebra p.102 This book states that "a bijective linear map from a vector space to another vector space is called an isomorphism". As far as know, generally isomorphism means ...
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### Can you define a vector space in terms of a pre-existing projective space?

Projective spaces are usually defined as the quotient of a vector space (by the equivalence relation that identifies collinear vectors). However, in my opinion, projective spaces seem intuitively less ...
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### About the definition of n-tuple

I've read from the theory of sets that the definition of ordered pair is foundational. To define formally the term of $n$-tuple I suppose we need to use the concept of ordered pair as well as the ...
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### Linear independence and dependence of vectors

I am really stuck in this problem, I have only 2 days to learn matrix's base, and its generator. My problem is that I know definitions but I don't understand intuitively what they mean. What I know: ...
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### Reason for existence of 'swapping' elementary matrix operation

In my Hoffman & Kunze book on Linear Algebra, we define 3 elementary row operations, the first of which is the swapping of two rows. I'm wondering why we need to even have such an elementary ...
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### Is a linear combination linearly independent?

I am a bit confused... Linear combination means $$F(X)=af(x_1)+bf(x_2) + \cdots$$ and linearly independent means $$af(x_1)+bf(x_2) + \cdots=0$$ where $a=b=\cdots=0$ My question: is a linear ...
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### Meaning of index in matrices

Question is, what does "index" mean? For systems of order greater than the number of characteristic roots of $C$ of index one Also, can anyone explain why is $u_1 + u_2 + n -1 =0$ and what ...
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### What is the intuitive meaning of the adjugate matrix?

The definition of the adjugate matrix is easy to understand, but I have never seen it used for anything. What is the intuitive meaning of this matrix? Are there examples of applications which may ...
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### Why are vector spaces sometimes called linear spaces?

I have never come across the term 'linear space' as a synonym for 'vector space' and it seems from the book I am using (Linear Algebra by Kostrikin and Manin) that the term linear space is more ...
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### Suppose I said “$X$ spans $W$”…

So I've seen two definitions of this: Let $V$ be a vector space with subspace $W$. We say that $X \subseteq V$ spans $W$ if and only if (Definition 1): Every $\vec{w} \in W$ can be written as ...
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### Question about piecewise linear paths

I'm having trouble with a concept about piecewise linear paths. I've been looking on the net but I haven't found a definition of it. Consider a finite family of hyperplanes in a finite-dimension real ...
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### What are the technical reasons that we must define vectors as “arrows” and carefully distinguish them from a point?

I thought I'd bring this question to math.SE, as it could spark some interesting discussion. When I first learned vectors - a long time ago and in high school - the textbook and teachers would always ...
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### Isotropic subspaces [duplicate]

Possible Duplicate: Etymology of the word “isotropic” Let $V$ be a vector space and we have a symmetric, non-degenerate bilinear form with signature $(n,n)$ on it. A subspace ...
Is it ok to assume that $\operatorname{dim}(\operatorname{Im}(T^*))=\operatorname{dim}[(\operatorname{Im}(T))^*]$, where $T$ is a linear map acting on a finite dimensional space. i.e. just taking the ...