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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$T$ : $R_{2×2}$ → $R$ with $T(A)$ = $det(A)$

Determine if the given function is a linear transformation and completely justify the answer. $T$ : $R_{2×2}$ → $R$ with $T(A)$ = $det(A)$
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13 views

Gaussian eliminatin of matrix $n\times n$

I am solving a problem and I got stuck on this - to find the row echelon form of this matrix: $$\begin{pmatrix} x_{11}&x_{21}&\cdots&x_{n1}\\ x_{12}&x_{22}&\cdots&x_{n2}\\ ...
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3answers
29 views

Confusion about how the determinant changes when all rows are multiplied by a scalar

I am having some trouble thinking about properties of the determinant. I understand why it is true that if $B$ is the matrix obtained from an $n \times n$ matrix $A$ by multiplying a row by a scalar ...
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1answer
18 views

Why does the unit vector of form $x_i=\frac{-1}{\sqrt{n}}$ minimize sum of $x_i$?

Cauchy-Schwarz implies that for $||\vec{x}||=1, \vec{y}=(1,\ldots,1)\in\mathbb R^n,\sum_{i=1}^{n} x_i = \pm\sqrt{n}$ if $\vec{x}=\pm{k}\vec{y}$. This implies that ...
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0answers
7 views

convertion into integer linear program

I am trying to model the Ising spin state problem into Integer linear program and find the optimal ground state using lp_solve. (This is just a miniature version of Ising state problem) $$ maximise: ...
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1answer
24 views

Can there exist a non-linear unitary map?

Suppose $V$ is a complex Hilbert space with inner product $( \cdot, \cdot)$. Can there exist a 'unitary function' $U : V \to V$ which is not linear? In other words, does $$(Uv, Uw) = (v,w) \ \ \text{ ...
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41 views

Linear algebra over a nonassociative division ring

In the question linear algebra over a division ring vs. over a field is discussed the relationship between linear algebra over a field and linear algebra over a division ring. Roughly speaking the ...
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33 views

What is the $\dim L(X,Y)$?

Let $X$ and $Y$ be two finite-dimensional vector spaces over the same field $K$, and let $L(X,Y)$ denote the vector space of all linear operators $T \colon X \to Y$. Then what is $\dim L(X,Y)$? My ...
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19 views

Eigenvalues of nilpotent matrices

I have these two claims for a real $k\times k$ matrix $A$ 1 If $A^n=0_{k\times k}$ for some $n\in\mathbb N$ and $\lambda$ is an eigenvalue of $A$, then $\lambda = 0$. 2 If $A^n=0_{k\times k}$ ...
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1answer
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Definition of Multiple .

Definition of multiple is : In mathematics, a multiple is the product of any quantity and an integer. In other words, for the quantities a and b, we say that b is a multiple of a if b = na for ...
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1answer
20 views

Every point in a simplex is a convex combination of p and a point in $C^{(p)}$

Let's fix an arbitrary point $p \in \Delta_n = \{(x_1, ..., x_n) \in \mathbb{R}^n \ : \ \sum_{i=1}^n x_i = 1 \}$ Could you help me prove that every point in a simplex $\Delta_n$ can be written as a ...
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3answers
29 views

Solve the Equation.

$$ \begin{bmatrix} 1 & 1 \\ 1 & 1 \\ \end{bmatrix} \begin{Bmatrix} v_1 \\ v_2 \\ \end{Bmatrix}= \begin{Bmatrix} 0 \\ 0 ...
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What are the possible topics for graduate dissertation with Mathematica? [on hold]

I am a graduate student. I am planning to do one year project work on linear algebra using Mathematica. Until now i have studied transformations,elementary canonical forms ,rational and jordan ...
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1answer
22 views

Having trouble finding all the equations for this circuit. Feel like i'm missing something crucial (Kirchoff law) [on hold]

Here's the question : http://imgur.com/9q4akZl [1] So, i divided the equations into loops and intersections. Loops: i1R+i2R+i4(0)=V i1R+i3R+i5(0) = V Intersections: i1-i2-i3 = 0 i2-i4 = 0 ...
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3answers
51 views

If M,N are finite dimensional vector spaces with same dimension ,then if M is subset of N ,then M=N

If M,N are finite dimensiona;l vector spaces with same dimension then if M is subset of N ,then M=N I think i need to show that both vector spaces are spanned by same bases in order to do this or to ...
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1answer
19 views

How to know what steps to perform when doing Gauss-Jordan reduction?

I am currently studying linear algebra at university. We are currently doing Inverses of matrices using Gauss-Jordan Reduction, but no one has really explained how you are supposed to decide what ERR ...
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24 views

Topics for master's project using Mathematica. [on hold]

I wish to do 1 year project in linear algebra using Mathematica. What are linear algebra topics which can be implemented on Mathematica as part of 1 year graduate level dissertation.
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1answer
11 views

How is multiplication defined on Grassman ring?

I'm reading Kenneth Hoffman's "Linear Algebra", Ed 2. In $\S5.7$ "the Grassman Ring" it briefly mentioned: The exterior product defines a multiplication product on forms and extend it linearly ...
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1answer
22 views

Finding the Standard Matrix of a Linear Transformation

The question is asking for Find the characteristic polynomial of the matrix: $$ A = \begin{bmatrix} 5 & 5 & 0 \\ 0 & 4 & -5 \\ -1 & 3 & 0 \\ ...
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2answers
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Prove you can choose orthonormal bases of any two subspaces of Euclidean space such that $(e_i, f_j)=0$ if $ i\neq j $

Prove you can choose orthonormal bases $(e_1,...,e_k)$ and $(f_1,...,f_j)$ of any two subspaces of Euclidean space such that $(e_i, f_j)=0$ if $i\neq j$ and $(e_i, f_j) \geq 0$ This is a question ...
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Proof that $(A+B)^T=A^T+B^T$? [on hold]

Homework question: Proof that $(A+B)^T=A^T+B^T$? Let $A$ and $B$ be $m\times n$ matrices. Prove that $(A+B)^T=A^T+B^T$ by comparing the ij-th entries of the matrices on each side of this equation. ...
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1answer
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Linear Algebra Determinant problem

I answered 18 assuming we could sub -3 into A. This was not the case, and I understood that. I checked the solution and it said that you factor out a 4! What?! I tried everything.. I even tried to ...
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2answers
42 views

Find orthonormal basis of $\mathbb{R}^3$ with a given span of two basis vectors

What is an orthonormal basis of $\mathbb{R}^3$ such that $\text{span }(\vec{u_1},\vec{u_2})=\left\{\begin{bmatrix}1\\2\\3\end{bmatrix},\begin{bmatrix}1\\1\\-1\end{bmatrix}\right\}$? I was thinking I ...
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0answers
24 views

Conditions for a Matrix to be Diagonalizable

Let $M$ be a matrix with the entries $a_{1}, ..., a_{n}$ on the secondary diagonal (the one that ranges from $m_{n1}$ to $m_{1n}$) with all other entries being $0$. Find under which conditions the ...
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0answers
30 views

Geometrical properties of tetrahedra under rotation

Consider two tetrahedra which share the same point of origin but differ in both scale and rotation over the X-axis. Can someone explain why the following points meet with these parameters? Both have ...
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0answers
10 views

stabilizable ,detectable and regulator

Assume that $(A_2,B_2)$ is stabilizable and $(C,A)$ is detectable then there exist aregulator if the equation $TA_1-A_2T-B_2V=A_3$ $D_1+D_2T+EV=0$ have solution (T,V).if $A_1$ is antisatable the ...
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0answers
71 views

Is Abstract Algebra really that difficult? [on hold]

I'm an undergraduate junior studying Mathematics, and I am planning on taking my Abstract Algebra sequence this following school year. I'm a little concerned because I have only taken Calculus 1-3, ...
2
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1answer
31 views

Rank of a symmetric matrix. (ISI Sample Paper)

Here, $\langle v,w\rangle=v^tw$ is the usual dot product. Let $A$ be an $n \times n$ symmetric matrix. Let $l_1, l_2, \ldots , l_{r+s}$ be $(r + s)$ linearly independent $n\times 1$ vectors such ...
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1answer
28 views

What is the upper bound on the error of the solution of a noise perturbed linear system $Mx=b$?

Let $x$ be solution to the following linear system: $$ Mx = b$$ and let $ \tilde{x}$ be the solution to the above linear system with some additive noise: $$ M \tilde{x}= \tilde{b}$$ where ...
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1answer
27 views

How to remove fields from sudoku puzzle in such way to assure there's still only 1 solution?

I'm trying to create a Sudoku puzzle (programatically, if that matters). Here's how I do it. STEP 1: Creating an initial set, with unique solution: 123456789 456789123 789123456 ...etc... STEP 2: ...
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1answer
47 views

Isometry of two Euclidean structures on the same vector bundle

I'm reading Milnor & Stasheff's Characteristic Classes, and I was struck by the following problem, which they call the Isometry Theorem: "Let $\mu$ and $\mu'$ be two different Euclidean metrics ...
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1answer
29 views

A and B are similar then adj(A) and adj(B) are similar [on hold]

If $A,B \in {M_n}$ are similar, why $adj(A)$ and $adj(B)$ are similar?
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2answers
25 views

Finding the Matrix of a Given Linear Transformation T with respect to a basis

I've been scouring the web and my textbook to no avail. I understand how to transform a matrix with respect to the standard basis, but not to a basis comprised of matrices. I need to find the matrix ...
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1answer
15 views

Minimal polynomial and diagonalization.

How would I be able to prove that a linear endomorphism $T$ is not diagonalizable if it's minimal polynomial is $x^2(x-1)(x+1)$? I thought I could try to put this into Jordan blocks and say that ...
2
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1answer
37 views

Finding solutions to a system of linear equations

I was working on this problem, to which I have the answer, but it is just that, an answer with no explanation, and I am stuck on how the answer was arrived at, for a few parts of this question. In ...
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1answer
27 views

Proving an $n\times n$ matrix is similar to its transpose

Is proving that an $n \times n$ diagonalizable matrix $A$ is similar to $A^T$ different that proving that $A\sim A^T$ when $A$ is not $n \times n$ and diagonalizable? I think I can work from ...
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2answers
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Why does $(\vec{v}\cdot\vec{u})\vec{u}\neq(\vec{u}\cdot\vec{u})\vec{v}$?

I'm trying to save time running Gram-Schmidt. Why doesn't this product equal $||\vec{u}||\vec{v}$? More specifically (and I know this is fundamental and that I should already know it), why doesn't the ...
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1answer
15 views

Apply gauss method to a linear system and them use results in another system

I have an exercise for my last assignment of linear algebra, which is the following: I tried to row reduce to echelon form the matrix created by the first linear system of equations and I obtain ...
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1answer
34 views

Find a vector whose image under $T$ is the vector $b$

Hey everyone, I'm having some trouble solving this problem. To find the vector can I multiply the matrix $A$ by the column vector $[a, b, c]$ and set that equal to the vector $b$? or do I have to ...
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1answer
17 views

Prove column space is a subspace of $\mathbb{R}^n$

I have an exercise on my last assignment for linear algebra, which is the following: The column space $C(A)$ of linear mapping $A: \mathbb{R}^m \rightarrow \mathbb{R}^n$ is defined by: ...
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Three dimensional rotation of equations.

I have a set of equations that describe a wire in (100) direction. I want to rotate the wire such that it's in the direction (111). My initial plan (which failed) was to use Euler coordinates and ...
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Proving the eigenvalues of a real symmetric matrix are real [duplicate]

Can I prove that, if $A$ is a real symmetric matrix, that its eigenvalues are real? Can I also prove that the eigenvectors associated with distinct eigenvalues of $A$ are orthogonal? I'm not sure ...
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6answers
118 views

Linear transformation that is like projection

I am given a linear transformation $\;T:V\to V\;$, with $\;V\;$ linear space over field $\;F\;$ , and with $\;\dim\text{Im}\,T=1\;$ . I am asked to prove that there exists scalar $\;c\in F\;$ such ...
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Is there any definition of such semi-bilinear?

$K$ is a finite field which not equal to its base field $F_2$. Let $f: V \rightarrow K$ be a function and $B(x,y)=f(x+y)+f(x)+f(y)$ such that $B(x+y,z)=B(x,z)+B(y,z)$ and $B(z,x+y)=B(z,x)+B(z,y)$ for ...
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2answers
32 views

Learning about convex optimisation

I'm interested in learning a bit about convex optimisation. The wikipedia article contains the following paragraph: The convexity of $f$ makes the powerful tools of convex analysis applicable. ...
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3answers
38 views

If $q\neq 0$ and $e$ are column vectors, when do we have $A$ such that $Aq=e$?

Question: if $q\neq 0$ and $e$ are $n\times 1$ column vectors, when do we have $A$ ($n\times n$) such that $Aq=e$? I've got a feeling that such $A$ always exists. Let $q_j$ be a nonzero element of ...
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1answer
20 views

Proving there is always a unique solution to $A^TAx = A^Tb$ and that $b - Ax$ is orthogonal to $R(A)$?

If $A$ is an $n \times k$ matrix of rank $k$ and $b \in R^n$, how can I prove there is always a unique solution to $A^TAx = A^Tb$ and that $b - Ax$ is orthogonal to $R(A)$? I know the dimension of ...
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0answers
10 views

Specify coordinate system from normal, azimuth and elevation angles.

I have a plane in $\mathbb R^3$, with a normal unit vector $\vec n=<n_x,n_y,n_z>$. The direction of the normal unit vector $\vec n$ is described from plane's altitude and azimuth angles $\alpha, ...
2
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1answer
14 views

Prove $U = \{u(x) \in P_4 | u(0) + u(1) = 0\}$ is a subspace of $P_4$

I am trying to prove $U = \{u(x) \in P_4 | u(0) + u(1) = 0\}$ is a subspace of $P_4$ For U is nonempty I have: Let $u(x) = 0x^4 + 0x^3 + 0x^2 + 0x + 0$ For U is closed under $+$ I have: Let $x, y ...
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2answers
51 views

Why the column space of a matrix is useful?

I know what is the column space of a matrix: it is basically the subspace formed by the linear combinations of the columns (vectors) of a matrix. From wikipedia, we have the following nice picture: ...