Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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Linear Algebra Question , to prove or to disprove

I need to pick out true statements from given below Let $ A $ be a $n \times n $ square matrix over $\mathbb R$ .Then pick out true statements from below: $1.$ There exists a real symmetric $n ...
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Finding a maximal isotropic subspace

I have the following question: Let $V$ be a finite dimensional complex vector space. For a given bilinear form $(,): V \times V \rightarrow \mathbb{C}$, a subspace $W$ of $V$ is called isotropic with ...
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41 views

A question about minors of matrices

Let $B_{\bar{i}\bar{i}}$ denote the remnant of a square matrix $B$ after its $i^{th}$ row and $i^{th}$ column have been removed. Now given any vector $v$ is there some natural relation between the ...
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236 views

diagonalizable matrix such that sum of every column is the same number

Let A be a square matrix which is diagonalizable over field $\mathbb{F}$ and the sum of the entries of any column is the same number $a\in\mathbb{F}$. Show that $a$ is eigenvalue of the matrix A. ...
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238 views

Can I Start Analysis? Seeking Your Advice on My Journey to Mathematics! [closed]

I am a college sophomore with double majors in mathematics and microbiology, and I have been doing independent research in the mathematical/computational biology, which really led me to love the ...
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59 views

Max - min problem of a quotient of norms

For the $2\times2$ matrix $\begin{bmatrix}4&0\\-3&-5\end{bmatrix}$ Part 1 Find nonzero vectors $u$ and $w$ that maximize and minimize respectively the quotient $||Av|| / ||v||$. Part 2 ...
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How to solve $xa=yb$ for $x,y$ in a ring

I'm probably missing something obvious, but does the equation $xa=yb$ necessarily have nontrivial solutions (where $x,y$ are not both zero) in a nonzero ring (i.e. $1\ne0$)? If one of them is zero, ...
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180 views

The meaning of notation like $f\colon \mathbb R^2 \to \mathbb R$, $x \in \mathbb R^n$, and $x \in \mathbb R$.

I am in second year university and am taking linear algebra this semester. Never having been a strong maths student, I am certainly struggling with some basic concepts and especially notation. I ...
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49 views

Can one state the idea of collinearity in linear algebra terms?

The elementary geometry book I have states: Points that lie on the same line are called collinear. In order for us to say that a point is between two other points, all three of the points must be ...
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18 views

Row reducing with unknown values

This is the first time I take linear algebra, so pardon my seemingly trivial question: How do I form zeroes under the leading entry if there is an unknown in the same column? \begin{matrix} k ...
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63 views

Computing an orthogonal projection,

I'm trying to find a vector in $\mathbb{R}^4$ that is both orthogonal to the space $W$ spanned by $\{(1,2,0,1), (0,1,1,1)\}$ and happens to be "closest" to the vector $(3,3,3,3)$. From reading ...
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129 views

Converting a Vector Equation into a Scalar Equation where the Direction Vector is $(1,0)$

I've been working on some textbook problems, where I've had to convert vector equations into scalar equations. I've understood it for the most part, by converting the vector equation into parametric ...
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40 views

How do I find the “surface normal” of a tetrahedron in 4 dimensions?

I am currently making a 4D game engine. It shows specific 3 dimensional cross-sections of the tetrahedrons that have been programmed in (in a 3 dimensional cross-section, each tetrahedron is seen as a ...
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1answer
63 views

jordan canonical form with direct product?

I met some problems when solving Jordan canonical forms. Here are two problems: Let $f: K^3\to K^3$ be a map in JCF having the matrix: $$\begin{pmatrix} -1 & 1 & 0\\ 0 &-1&1\\ ...
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1answer
77 views

Proving that given functionals span the dual space of all polynomials of 4-th degree (and finding basis for which given functionals are dual basis)

Let's call $\Bbb{R[x]}_4$ a linear space of all polynomials of 4-th degree with real coefficients. We are given functionals $\phi_j \in (\Bbb{R[x]}_4)^*$: $\phi_0(p)=p(-1)$ ...
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Asking for some reference for square of a graph Laplacian matrix. ($L^2$ or $L^{\dagger^2}$)

I am looking for some information regarding Laplacian squared of a graph. ($L^2 or L^{\dagger^2}$) I couldn't find anything special. Specially the graphs with positive weights on edges. Any related ...
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92 views

prove that if U is a subspace of the finite dimensional vector space V such that dimU=dimV then U=V

prove that if U is a subspace of the finite dimensional vector space V such that dimU=dimV then U=V I try with the following let dim U = dim V = n. ∴thhere are n independent vectors u1, u2, u3, ...
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42 views

Relationship between $[T]_{\mathcal{B}}$ and $[T]_{\mathcal{B'}}$.

Let $T$ be a linear operator on a finite-dimensional space $V$. Let $\mathcal{B}=\{\alpha_1, \dots, \alpha_n \}$ and $\mathcal{B'} = \{\alpha'_1, \dots, \alpha'_n\}$ be two basis for $V$. How are ...
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60 views

Why is $\langle p,q \rangle = 0$? [duplicate]

First of all, sorry for opening a new question about it, but I'm curious to understand: John Hughes claims that $\langle p,q \rangle =0$ (in the end of his answer) Why is it true? Prove that there ...
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1answer
28 views

Condition for existence of a unique solution for a desired variable in a system of linear equations

Consider a system of linear equations of the form $$\mathbf{A}\mathbf{x}=\mathbf{b}, \mathbf{A} \in \mathbb{R}^{L\times K}, \mathbf{x} \in \mathbb{R}^{L}, \mathbf{b} \in \mathbb{R}^{K} $$ with $L$ ...
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50 views

Schur's Triangularization Lemma in Hefferon's Linear Algebra textbook

I'm reviewing some material and came to this: Fix a basis $B = \{\vec{\beta}_1, \ldots, \vec{\beta}_n\}$ for $V$ ($V$ is a vector space) and observe that the spans $$ [\emptyset] = \{\vec{0}\} ...
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118 views

Is a normal matrix satisfying $A^TA=…$ circulant?

Let $A=\{a_{ij}\}$ be a normal matrix such that $a_{ij}\geq 0$ with equality iff $i=j$. Suppose that $$ A^TA=\begin{pmatrix} a & b & \cdots & b\\ b & a & \ddots & \vdots\\ ...
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If $A$ is an invertible matrix, show that $\det(A)$ not equal $0$ and $\det(A^{-1})$ not equal $0$?

If $A$ is an invertible matrix, show that $\det(A)$ not equal $0$ and $\det(A^{-1})$ not equal $0$. My attempt to solve it but I am not sure it is correct: Given $A$ is an invertible matrix. ...
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75 views

Prove that there is a unique inner product on $V$

Let $V$, a vector space over $\mathbb{F}$ and $W_1,W_2 \subseteq V$ such that $W_1 \oplus W_2 = V$. For $i=1,2$, let $\langle , \rangle $ on $W_i$. Prove that there is a unique inner product on $V$ ...
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37 views

problem with derivation of PCA as minimizer of MSE

I am trying to learn proof of the Principal Component Analysis(PCA) as the minimizer of mean square error between all orthonormal bases in M dimensional space. in part of proof we see that: Now we ...
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3answers
58 views

Why if matrix $A$ is invertible and $A(\mathbf{x-y})=0$ then $\mathbf{x-y}=0$?

When browsing through my algebra textbook on examples of isomorhic linear transformation in one of the proofs there was a statement that if matrix $A$ is invertible and for vectors $\mathbf{x,y}$ it ...
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2answers
87 views

Isomorphism of $(\mathbb{R^3},\times)$ [closed]

I am studying Lie algebras and I have difficulties doing this exercise: Show that $\mathbb{R^3}$ with vector product is isomorphic to the space of antisymmetric $3\times3$ matrices with operation ...
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29 views

Proving the unitary relation of ensemble decompositions

I apologize if this is better suited for physics.stackexchange; looking at previously asked questions it seems as if this would be a good fit here, as the arguments are mostly mathematical. In my ...
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46 views

Prove/Disprove: $\forall u\in V: \langle v,u \rangle = \langle w,u \rangle \implies v=w$.

Prove/Disprove: $\forall u\in V: \langle v,u \rangle = \langle w,u \rangle \implies v=w$. I want to say "Yes", but couldn't formulate my intuition into a proof. How to prove it?
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1answer
62 views

Multilinear algebra some basics.

The wedge product of $p$ vectors in vector space $V$ is called a $p$-vector and the vector space generated by all $p$-vectors is denoted $\bigwedge^p V$ with the basis $e_I:=e_{i1}\wedge\dots\wedge ...
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62 views

How to define an affine transformation using 2 triangles?

I have $2$ triangles ($6$ dots) is a $2D$ plane. The points of the triangles are: a, b, c and x, y, z I would like to find a ...
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2answers
32 views

How many ordered bases can I find?

I have found a basis for this question, however I am curious how many correct solutions I could find? Can you explain how I would calculate this?
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4answers
133 views

Prove that $A^3-3A^2+4A-5=0$ for a given matrix $A$.

Consider the following matrix $A$: $$A= \begin{bmatrix} 0 &1 &-1\\ 1 &1 &1\\ -1 &0 &2\\ \end{bmatrix} $$ $$\text{Prove that }\;A^3-3A^2+4A-5=0$$ I have no idea how to solve ...
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1answer
153 views

Closest point on a 3D triangle, is this algorithm correct?

Given a point $P$ and three triangle vertices $U$, $V$, $W$, all in $\mathbb{R}^3$, I need to find the point in the triangle $UVW$closest to $P$. Does the following algorithm work, or have I missed ...
6
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123 views

Trace of a matrix $A$ with $A^2=I$

Let $A$ be a complex-value square matrix with $A^2=I$ identity. Then is the trace of $A$ a real value?
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Finding the eigenvalues and eigenvectors of tridiagonal matrix

Let $A$ be a tridiagonal matrix as below: $${A_{n \times n}} = \left[ {\begin{array}{*{20}{c}} {a}&{b_1}&{}&{}&{}\\ {c_1}&{a}&{b_2}&{}&{}\\ ...
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3answers
171 views

Deriving inverse matrix formula

If matrix $A$ is given with dimensions $2 \times2 $ then, A is invertible if, and only, if $ad - bc \neq 0$: $$\begin{bmatrix}a & b\ \\c & d \ \end{bmatrix}^{-1} = \frac{1}{ad - ...
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1answer
126 views

How to express the Pythons' NumPy linspace or arange arrays mathematically?

How one can express digital one dimensional array, such as x = np.linspace(0, 10, 1000) or x = np.arange(-1, 1, 0.01) (examples ...
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35 views

Good and thorough online and/or free Matroid Theory references?

I'm studying a course on Matroid theory. Sadly, I can't really afford buying the textbook, so I only use the lecture notes, which aren't enough for me. Are there any good and thorough online and ...
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1answer
85 views

The subspace sum of a point and a closed subspace is closed

In projective geometry, a polarity is a map $\ell\mapsto\ell^\perp$ on the subspaces of $\Bbb P$ satisfying the axioms: $\Bbb P^\perp=0$ $\ell\subseteq m\implies m^\perp\subseteq\ell^\perp$ If $P$ ...
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Problem with Tangent of an Angle

I have a setup with two vectors. $\vec x = (10,0,0)$ and $\vec y = (10,3,0)$ By rotating around the xy origin I can create a projected circle of radius 3. I wanted to find the angle between $\vec x$ ...
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38 views

What is the solution set of the given homogenous system?

Write the solution set of the given homogenous system in parametric vector form: \begin{align} 2x_{1}+2x_{2}+4x_{3} &= 0\\ -4x_{1}-4x_{2}-8x_{3} &= 0\\ 0x_{1}-3x_{2}-3x_{3} &= 0\\ ...
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What are some Applications of the Permanent of a Matrix?

I have a decent understanding of the determinant of a matrix in terms of its role in Telling you if a matrix is invertible (zero vs. nonzero) Expressing the product of a matrix's eigenvalues with ...
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1answer
19 views

Question regarding cyclic subspaces

Let $T:V\to V$ be a linear operator on a finite dimensional field and let $\vec v \in V$ ($\vec v \neq \vec 0$) such that $\operatorname{Span}${$\vec v, T(\vec v), T^2(\vec v),...$}$\neq V$ If $\vec ...
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Is there a generalization of the Lagrange polynomial to 3D?

What is a way to construct a smooth polynomial surface ($\mathbb{R}^2 \rightarrow \mathbb{R}$) with Lagrange-polynomial properties in every partial derivative? I want to try this for image ...
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Dimension of generalized eigenspace.

Let $T \in \text{Hom}_F(V,V)$, suppose the characteristic polynomial of $T$, $c_T(x) = (x- \lambda)^kp(x)$, where $p(\lambda) \neq 0$, show that $\text{dim}_F (E_{\lambda}^\infty) = k$, where ...
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Orthogonal complement propertiss

I'm required to proove that $W^{\perp} + U^{\perp} \subseteq (W \cap U)^{\perp}$. I've already proven $U \subseteq W \to W^{\perp} \subseteq U^{\perp}$ and $(U+W)^{\perp} = W^{\perp} \cap U^{\perp}$. ...
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49 views

Matrix of linear operator

I have troubles with following problem, can you please help me? Matrix of linear operator $f$ over the field $\mathbf{Z}_5^3$ to a canonical base is $A$. $$A= \left( \begin{array}{ccc} 3 & 1 ...
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86 views

Generic method to find a matrix whose null space is given

I was given a null space of a matrix: $$ N(A) = \operatorname{sp}\left\{\begin{pmatrix} 1 \\ 0 \\ 1 \\ \end{pmatrix}, \begin{pmatrix} 2 \\ -1 \\ 1 \\ \end{pmatrix} \right\}. $$ I found the following ...
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Conjugate diagonal matrices

Let $M$ be the set of matrices that have precisely one entry in each row/column that is nonzero and $D$ be the set of invertible diagonal matrices ( so all entries down the diagonal are nonzero). ...