Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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3D vertices $\rightarrow$ 2D polygon $\rightarrow$ 3d transformation

This may be a little strange. I have an array of 3D vertices, which represent a 3D face (n-gon). I first need to describe the face in 2D. I can then apply these transforms on it. X, Y, and Z ...
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50 views

Question about reduction of spanning set

Consider theorem 5 here. It is the statement If $v_1,...,v_n$ span $V$ then they may be reduced to a basis of $V$. I am wondering about why the proof proceeds with a distinguished step one. Can ...
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114 views

Operator Norm of $ 3 \times 3$ matrix

Find the operator norm $\|A\|_o = \sup \|Ax\|_2$, the supremum being taken over all $x \in \mathbb{R}^3$ with $\|x\|_2 = 1$. Here $A$ is the matrix $$\begin{pmatrix} 0&1&1\\ 1&0&1 \\ ...
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Prove:$x^d-1 \mid x^n-1$ iff $d \mid n$.

Prove:$x^d-1 \mid x^n-1$ iff $d \mid n$. my tries: necessity, Let $n=d t+r$, $0\le r<d$ since $x^d-1 \mid x^n-1$, so, $x^n-1=\left(x^d-1\right)\left(x^{\text{dt}+r-d}+\dots+1\right)$... ...
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307 views

Eigenvalue of linear operator iff eigenvalue of matrix representation.

I'm trying to prove the following theorem, which seems straightforward enough, but I'm confused about the wording and proving the converse: Let T be a linear operator on a finite-dimensional vector ...
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81 views

Two orthonormal bases

Suppose that $\{u_1, \dots, u_n\}$ and $\{v_1, \dots, v_n\}$ are orthonormal bases. Define the $n \times n$ matrix \begin{equation} A = \sum\limits_{j=1}^n \mu_j v_j \otimes u^t_j, \end{equation} ...
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88 views

Solutions to $AX=B$

Is my proof of the following statement correct? Claim. Let $M^\prime=[A|B]$ be a reduced row echelon matrix. Then $AX=B$ has solutions iff there is no pivot in the last column $B$. My proof. If ...
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101 views

Finding the jordan form of a funny looking matrix

I was working on this problem from a previous qual exam. Just when I thought I knew how to find the Jordan form of any matrix and then I find this....aaaargh :) Find the jordan form of the ...
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How to find the best fit when you have a set of ideal ratios, but some of those are below a minimum?

Say you have a set of ideal ratios, whose sum = 1. For example, input = [0.2, 0.2, 0.3, 0.3] But suppose that there is a rule stating that every ratio should be ...
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89 views

Set of all matrix of rank $ r $ is open set in $ M_n (\mathbb { R })$

I have no idea how to start it. Actually I have no idea which matrix in $ M_n (\mathbb {R})$ are of rank $ r $. I know all basic result about it. please help me.
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678 views

Eigenvalues of normal matrix

I want to show that $\lambda$ is an eigenvalue of a normal matrix $A$ if and only if $\overline{\lambda}$ is an eigenvalue of $A^{*}$. I am trying to show it for a while and I guess there are some ...
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44 views

Prove that the function T is a projection.

Let $T:V\rightarrow V$ be a projection on the vector space $V$. Prove that: $I - T$ is a projection $V = \ker(I-T) \oplus \mathrm{im}(I-T)$ How do I show that $(I - T)^2 = (I - T)$. I think $I$ ...
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175 views

Doubt on determinant and linear independence

I am confused about this matrix. We know Row rank = column rank = determinant rank for a matrix and proof is known to all. See the following matrix $$A = \left[\begin{array} &t &t^2\\ 0 ...
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613 views

Proof related to direct sum and subspaces

I did the following exercise: If $U_1, U_2, W$ are subspaces of $V$ with the property $V = U_1 \oplus W = U_2 \oplus W$ then $U_1 = U_2$. My proof: Assume $u_1 \in U_1 \subset V$. Then $u_1 \in U_2 ...
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56 views

Example that is not a subspace

I did some linear algebra exercise and did the following: Give an example of a nonempty subset $U$ of the xy-plane with the property that $U$ is closed with respect to scalar multiplication but $U$ is ...
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55 views

Show that $V$ is $T$-invariant.

Let $T:W \to W$ be a linear operator vector space $W$ over $\mathbb{F}$. such that $w \in W$ where $$\{w, T(w) ,T^2(w)\}$$ is linearly independent and $T^3(w)= w +T(w)+T^2(w)$. Show that $$V := ...
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$A$ complex, not diagonal, can $A^*A$ be diagonal?

If $A$ is a complex matrix which is not diagonal, can $A^*A$ be diagonal? My first impression is that it cannot, and my mind runs to the fact that $\operatorname{Tr}{(A^* A)}\geq \sum ...
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86 views

Proof by example?

I was doing some linear algebra exercies. One I did is this: Prove that the union of two subspaces $U,W$ of $V$ is a subspace of $V$ if and only if one contains the other. My proof is this: If one ...
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96 views

on the Singular Value Decomposition

If $T$ is a self-adjoint linear map on a $n$-dimensional inner product space $X$ (either real or complex) then we know by the spectral theorem that there is an orthonormal basis of $X$, call it $v = ...
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190 views

Positive semi-definite matrix problem

If $A,B$ and $M$ are positive semi-definite matrices, and we have $$ A+B \succeq M .$$ Do there always exist two positive semi-definite matrices $ M_{1}, M_{2} , $ such that $$ A \succeq M_{1}, ...
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645 views

What is the mathematical intuition behind àl-jàbrà?

The term algebra comes from the arabic term àl-jàbrà that means "to force", "to restore". Over centuries mathematicians, in east and west, celebrate by this term mathematical disciplines. What is ...
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154 views

Iwasawa Decomposition

I was asked to prove that if $$ T_{n}^{+}(\mathbb{R}) \subseteq M_{n}(\mathbb{R})$$ denotes the set of upper triangular matrices with positive diagonal entries, then prove that the multiplication ...
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Compute $(-1)^n\sum_{k=1}^n (-1)^k\frac{(k+n-1)!}{(k-1)!(k-1)!(n-k)!}$

Compute $(-1)^n\sum_{k=1}^n (-1)^k\frac{(k+n-1)!}{(k-1)!(k-1)!(n-k)!}$ Define $a_{k,m}=\frac{(-1)^{k+m}(n+k-1)!(n+m-1)!}{(k+m-1)[(k-1)!(m-1)!]^2(n-m)!(n-k)!}$ Compute ...
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119 views

Is this a valid proof of why a linearly dependent matrix has a $0$ determinant?

If a $3\times{3}$ matrix has a non-trivial solution to $B\vec{x}=\vec{0}$, then it has linearly dependent rows. Looking at Cramer's rule: $$x_1=\frac{\begin{vmatrix}0 & b_{12} & b_{13} \\0 ...
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899 views

Why is $\det (A-\lambda I)=0$?

I'm not sure I understand the logic behind why $\det (A-\lambda I)=0$ for any non-trivial solution to $(A-\lambda I)x=0$.
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50 views

Determinant of symmetric matrix of the form $v\otimes v$

Note that for $V=\mathbf{R}^n$, $$S^2V = \{ v\otimes w \mid v, w\in V\text{ and }v\otimes w=w\otimes v \} =\{ A\in \mathrm{M}_2(\mathbf{R}) \mid A=A^T \}.$$ Clearly, $S^2V $ contains $O=\{ v\otimes v ...
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156 views

Ways to calculate the inverse of a matrix, assuming it exists…

I'm wondering - Other than by using row reduction on the augmented $[A|I]$ to get $[I|A^{-1}]$, and by reducing a matrix to a product of elementary matrices, is there any other way to determine what ...
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167 views

Vector space bases without axiom of choice

I want to find an example of a vector space with no base if we assume that axiom of choice is incorrect. This question might be duplicate so please alert me. Thanks.
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59 views

Exhibiting A Basis

Exhibit a basis and calculate the dimension of the following subspace S of $\mathbb{P}_2$. $S=\{a+b(x+x^2) \mid a,b \in \mathbb{R}\}$ $\mathbb{P}_n$ denotes the vector space with polynomials ...
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332 views

Closed Form for Continuant (Determinant Tridiagonal Matrix)

Consider the particular tridiagonal, $n \times n$ matrix $A$: A = $\left(\begin{array}{ccccccc} a_1&b_2 ...
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92 views

How to check linear independence of these vectors?

Let $R=k\langle x,y\rangle$ be the free algebra on two variables. We can think of it as an algebra of non-commutative polynomials. Consider elements of the form $p[x,y]q$, where $p,q\in R$ are some ...
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If $T: V \to W$ is an isomorphism, then there are bases $B$ and $B'$ such that $[T]_{B',B}$ is the identity matrix

Suppose $V$ and $W$ are finite-dimensional vector spaces and that $T: V \to W$ is an isomorphism. Then there exist bases $B$ and $B'$ for $V$ and $W$ respectively such that $[T]_{B',B}$ is the ...
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172 views

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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988 views

Hyperplanes and intersection

How to prove that if I have two hyperplanes in $\mathbb{R}^{n}$ that have only one point of intersection, then $n=2$ (or $n=1$ trivially).
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471 views

What's a good book on advanced linear algebra?

I'm taking an advanced linear algebra course and I'm a little confused about books. The teacher said we could use any book we wanted to, but he recomended just Hoffman and Kunze and also Kostrikin, ...
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Linearity of the determinant

I'd like to prove the following properties of the determinant map. $\det I = 1$ $\det$ is linear in the rows of the input matrix The determinant map is defined on $n\times n$ matrices $A$ by: ...
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1answer
31 views

Is the dual of an equivariant metric equivariant?

Let $g$ a finite dimensional $K$-vector space, and let $g:V \otimes V \to K$ be an inner-product. If As usual, we can use the musical isomorphisms of $g$ to define an inner product on $V^*$, which we ...
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3answers
509 views

Endomorphisms of a finite dimensional vector space

From Humphreys' Introduction to Lie Algebras and Representation Theory: If $V$ is a finite dimensional vector space over $F$, denote by $\text{End }V$ the set of linear transformations ...
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1answer
131 views

LU factorization with pivot to solve linear system

I read that LUP matrix exist for any square matrix such that it is not a singular one. But I came across a matrix that when I get the LUP and calculate Lz=Pb -> ...
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1answer
65 views

If $Af(B) = B$ and the constant term of $f$ is nonzero, $f(B)$ is invertible

I am trying to solve the following problem: Let $A,B \in M_n(\mathbb C)$ be matrices and $f\in \mathbb C[X]$ such that $Af(B) = B$. Prove that if $f(B)$ is not invertible, $f(0)=0$. I set out ...
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Proving a Set is NOT a vector space

Before I begin, I will emphasis I DO NOT want the full solution. I just want some hints. Show that the set $S=\{\textbf{x}\in \mathbb{R}^3: x_{1} \leq 0$ and $x_{2}\geq 0 \}$ with the usual rules for ...
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3answers
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Are parallel vectors always scalar multiple of each others?

I read this in a tutorial of a university course : We note that the vectors V, cV are parallel, and conversely, if two vectors are parallel (that is, they have the same direction), then one is ...
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60 views

How is row elimination getting rid of this entry?

This is a really elementary question, but I want to make sure I'm not missing something conceptual. In Strang's book Linear Algebra and Its Applications, on p. 321, he introduces tests for positive ...
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Are there open problems in Linear Algebra?

I'm reading some stuff about algebraic K-theory, which can be regarded as a "generalization" of linear algebra, because we want to use the same tools like in linear algebra in module theory. There ...
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1answer
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Does a “typical” unitary matrix have an entry of magnitude 1?

I guess that a "typical" unitary matrix (or "almost every" unitary matrix) in $d \geq 2$ dimensions does not have an entry with magnitude 1. I would like to make this statement more precise and see a ...
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40 views

linear map property.

I found the following theorem in "Friedberg-Lienar algebra 4ed". " Let $~~V,~W~$ be vector space over field F. Let $~ \varphi ~: V \to W ~~$ be $~~$ isomorphism . Then, For any $~Q \subset V$ ...
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110 views

Images in a short exact sequence

Suppose $$ 0\to V\to W\to X\to 0\\ \downarrow\quad\quad\downarrow\quad\quad\downarrow\\ 0\to V'\to W'\to X'\to 0\\ $$ is a commutative diagram of vector spaces, with the top and bottom rows short ...
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getting PDF from a given Moment Generating Function

if the moment generating function mgf of a random variable w is M(t)=(1-7t)-20 find the i)pdf ii)mean iii)variance of w
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82 views

The generalized eigenvectors of linear operator $T$ span space $V$, why?

I'm studying about determinant and I have a problem understanding the following (Proposition 3.4): The problems I have are highlighted with red rectangles. If anyone can, could you clarify these ...
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224 views

How to prove that a matrix $U$ is unitary, if and only if the columns form an orthonormal basis?

And also, is it true that a matrix is unitary if and only if $T^{-1}=T^{*}$ ? Thanks.