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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Simple proof that a 3x3-matrix with entries s and s+1 cannot have determinant $\pm$1, if s>1.

Let s>1 and A be a 3x3-matrix with entries s or s+1. Then $det(A)\ne \pm 1$. The determinant has the form as+b with integers a,b and it has to be proven that $a>0$ if and only if $b>0$ and ...
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21 views

Determinant of a matrix shifted by m

Let $A$ be an $n\times n$ matrix and $Z$ be the $n\times n$ matrix, whose entries are all $m$. Let $S$ be the sum of all the adjoints of $A$. Then my conjecture is $\det(A+Z)=\det(A)+Sm$ , in ...
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How are signs on eigen vectors chosen, am confused? Linear Algebra

I have found the eigen vaues, I also know that you can find the eigenvectors through a Gausian Jordan. For vector x1, gauss jordan gives me rows(1 -1/3 ,, 0 0 ) For vector x2, ...
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15 views

Geometrical interpretation of a solution of a system of linear equations with complex coefficients.

What is the geometrical interpretation of a (unique/infinitely many/no) solution of a system of linear equations of three(or less) equations with three(or less) unknowns with complex coefficients?
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26 views

Intersection between 2 lines (3D). This doesn't have a solution does it?

so I was looking through an old exam and this question was given: The teachers answer was the point (9, -9, 21) I tried solving this myself, I got x = x, y = y, but I could not find a point where ...
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1answer
27 views

Relation on the determinant of a matrix and the product of its diagonal entries?

Let $A$ be a $3\times 3$ symmetric matrix, with three real eigenvalues $\lambda_1,\lambda_2,\lambda_3$, and diagonal entries $a_1,a_2,a_3$, is it true that \begin{equation*} \det ...
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1answer
25 views

Linear Algebra, Inner product space, Normal operator question

I have solved a question but I am not sure the last step of the question. If someone can verify it that would be great. Let $V$ be a finite dimensional vector space with complex inner product. Let ...
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15 views

How to find an equation of a plane perpendicular to two other planes and passing through a point

Please, could anybody help me with the next problem. I have two planes: $$ 2x-y+5z+3=0 \ (\text{red plane})\\ x+3y-z-7=0 \ (\text{green plane}) $$ And I need to find a plane which is perpendicular ...
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38 views

Proving boundedness of a function .

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
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10 views

Sequences length for LFSR in the general case

An LFSR with a reducible polynomial can generate several sequences, depending on the initial value. My goal is to have an algorithm to compute those length without going through the enumeration of all ...
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56 views

Square root of symmetric matrix

I have a symmetric matrix $A$. How do I compute a matrix $B$ such that $B^tB=A$ where $B^t$ is the transpose of $B$. I cannot figure out if this is at all related to the square root of $A$. How to ...
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40 views

Is the set of all Taylor polynomials a vector space?

Let $V$ denote the set of all Taylor polynomials of degree $\leq n$ for a fixed natural number $n$ (including the zero polynomial), regraded as real-valued functions of a real variable. Then is $V$ a ...
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76 views

Automorphisms of $\mathbb{Z}/p\oplus\cdots\oplus \mathbb{Z}/p$

Consider the abelian group $$G = \underbrace{\mathbb{Z}/p\oplus\cdots\oplus \mathbb{Z}/p}_{n},$$ where $p$ is prime and $1\le n \le p$. I want to show that $G$ has no automorphism of order $p^2$. I ...
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20 views

Finding alternating series for Power series

Given data and conditions I have a power series, $PS(x) = \sum_{n=0}^\infty R_nx^n$. I have a infinite GP,something like G(x) = $\sum_{k=0}^\infty ax^k = \frac{a}{1-x} $ . Never take G(x),such ...
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1answer
28 views

Linear Transforms & Matrices

$T:R^4 -> R^3$ Linear Transform This matrix is $[T]_{B2}^{B1}$ = A =\begin{pmatrix}1&2&3&4\\1&4&0&2\\2&2&9&10\end{pmatrix} After elimination we get: ...
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What is the solution to the system $\frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1}$?

I'm trying to solve the system $$ \begin{matrix} & \frac{df_1}{dt} = kf_1+lf_2 \\ & \vdots \\ & \frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1} \\ & \vdots \\ & \frac{df_N}{dt} = ...
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1answer
37 views

Why is there no $[X,[X,[X,Y]]]$ and $[Y,[Y,[X,Y]]]$ in the fourth order term of BCH formula?

While trying to deal with a problem involving BCH (Baker-Campbell-Hausdorff) formula, I've noticed something strange. Everywhere in the literature I've managed to fetch (for example: this and this ...
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2answers
55 views

The meaning of Inverse Matrix

I am studying Linear Algebra, I have 3 questions in my mind What does an inverse matrix mean. I am trying to have a meaning of it, but I don't really understand. When a matrix does not have an ...
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3answers
27 views

number of solution to the given equation.

a,b,c, are all non-negative integers such that a + b + c=100 and 1000a + 300b + 50c = 10000 How many such triplets are possible? i have tried to reduce ...
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1answer
30 views

Finding the area of a triangle from vertices? Linear Algebra

I pretty much did this problem, but I failed to get the few last blanks where they ask the area. Its confusing, they say its half the volume of matrix (u v w) in the start of the question. which means ...
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27 views

Orthogonal Complement, Direct Sum

Can someone please explain the proof of the theorem below? I've been looking at it for hours and couldn't figure out how to prove it. Thanks! Suppose $U$ is a finite-dimensional subspace of $V$. ...
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27 views

proof that T^k is a positive operator

so the book (Axler linear algebra done right) asks me to prove that if $T$ is a positive operator then $T^k$ is also positive , now the book defines a positive operator as an operator which is self ...
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1answer
61 views

Is the following Eigenvalue inequality holds or not?

Can anyone help me with the following problem? Suppose $u=(u_1,u_2,...u_n)^T$, $e=(1,1,...1)^T$, and we have $u\geq e$. Now for any symmetric matrix $A\in S^n$ with $diag(A)=0$, can we claim the ...
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41 views

The geometric multiplicity

By given this matrix: \begin{pmatrix}0&a&0\\0&0&1\\0&0&0\end{pmatrix} Why for any a which is not 0 the geometric multiplicity = 1? and why for a = 0 the g.m. = 2? I don't ...
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1answer
20 views

Matrix of a linear map Questions

Suppose n < m. Show there exists a basis $w_1...w_m$ of w for every choice of basis for v of degree n such that the last m-n rows of M(T) consist of only $0$'s for every choice of basis for w. ...
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61 views

Does adding linearly independent vectors retain linear independence?

Suppose the vectors u, v, w are linearly independent and u'=u+v, v'=v+w and w'=u+w. I'd like to check if u', v', w' are also linearly independent. I know they can be linearly independent, such as if ...
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2answers
62 views

What is a bilinear form?

I'm a CS master student and I'm reading a paper that mentions the term "bilinear form". Actually the paper mentions "bilinear regression model". But I think in order to understand what a "bilinear ...
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46 views

Linear Algebra, Eigenvalues and Eigenvectors Exercise

I have a question from an exercise. I am given a vector space over the field $\mathbb{R}^{3}$ with 2 dimensions and I am asked to find a basis of eigenvectors. I found the eigenvalues but I have ...
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1answer
25 views

Geometric meaning of a matrix decomposed into its symmetric and skew-symmetric parts

What's the geometric meaning of a matrix decomposed into its symmetric and skew-symmetric parts? For example, a skew-symmetric matrix on its own can be interpreted as an infinitesimal rotation. As ...
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Show that a matrix with monomial entries is invertible [duplicate]

Let $z_1,z_2,...,z_{n+1}$ be distinct non-zero real numbers. Show that ...
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1answer
42 views

Can I always extend an inner product from a real to a complex vector space?

Let V be a vector space over the real numbers with finite dimension. Let <,> : VxV -> R be an inner product on V. Let W be the same vector space V, but now considered as a vector space over the ...
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44 views

Proving boundedness of a function (part 1).

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
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0answers
34 views

What is the difference between the span of a set to its subspace?

I am confused with some of the definitions of linear algebra. I know that the span of set S is basically the set of all the linear combinations of the vectors in S. The subspace of the set S is the ...
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1answer
27 views

Finding equation of line with given slope

Find the distances of the point (1,2) from a straight line. The slope is given to be 5 and the line passes through the intersection point of the lines $x+2y = 5$ and $x - 3y = 7$ Obviously I could ...
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Set and List distinction - Golan Linear Algebra book

He says: A finite or countably-infinite selection of elements of a set A is a list. then he says: Note that the elements of a list need not be distinct: Taking the case of a list with ...
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87 views

If A+tB is nilpotent for n+1 distinct values of t, then A and B are nilpotent.

Suppose A and B are $n\times n$ matrices over $\mathbb{R}$ such that for n+1 distinct $t \in \mathbb{R}$, the matrix A+tB is nilpotent. Prove that A and B are nilpotent. What I've tried so far: ...
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1answer
27 views

Can unitary matrices be defined for infinite-dimensional space?

As the title says, can unitary matrices be defined for infinite-dimensional space?
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29 views

Vector (scalar) product: associativity

Let $x$, $y$, $z$ be vectors of $\mathbb{R}^{n\times1}$. Consider this scalar result: $b = x^{\top} y z$. The issue is that the above product does not follow the classical associativity algerbra ...
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proving a fraction with 2 parameters to be small

Hi I have a fraction as below $$\frac{1.623x^4+0.434x^4\sum_iy_iz_i^2+(0.014x^2+0.0027)\sum_iy_iz_i^4}{1.645x^2+(0.083-0.329x^2+0.435x^4)\sum_iy_iz_i^2+0.014\sum_iy_iz_i^4}$$ where $x\in[0, 0.5]$, ...
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1answer
19 views

System of linear algebraic equations in maple

I have to solve a system of linear algebraic equations in maple. In my book it is given using linalg package in maple, which is deprecated. So I want to use Linsolve from LinearAlgebra package, but ...
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1answer
61 views

solving the equation

let there be a function $ f(x)= \ln x-kx^2, k>0$ determine for whihc values of $ k$ ,the equation $f(x)=0.5$ has a single solution; attemp to solve: $$0.5 = \ln x-kx^2$$ $$kx^2 +0.5 = \ln x $$ ...
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1answer
21 views

Volume of a polytope cut off by a hyperplane

Given a maximization problem with constraints, and adding a few more constraints using the Gomory cuts and solving the relaxed maximization problem, we can arrive at integer solutions. I am looking to ...
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1answer
17 views

Rank of a simple matrix series

Problem Specifications and Given conditions I have a matrix $L$ with rank 3 and dimension $ 3 \times 3$. $L = K_0+\sum_{n=1}^{\infty}K_i $ . Rank of $K_0$ is 3 and rank of L is also 3. Rank of ...
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1answer
19 views

Matrix Rank calculation

I have a matrix A . A can be written as A=B+D. I know rank of B. It is 3. Is it possible for A to have ranks $<3$ . If so please prove.
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1answer
37 views

Rank of a Matrix Sum

I have matrices of $3\times3$ dimension such that S=A+B. I know there is one inequality connecting rank of the matrices A,B and its sum S? Could you write down that here. It will be a great help for ...
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10 views

conversion by schur or svd decomposition?

Inorder to perform eigen decomposition, I converted a rectangular matrix to square by multiplying with the transpose of the matrix. After decomposition, I got the component matrices. If I multiply the ...
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1answer
70 views

Project sin(x) onto orthonormal basis that span ${(1, x, x^2, x^3, x^4, x^5)}$ on domain $[-\pi, \pi]$

I am self-studying LA through Linear Algebra Done Right 2nd ed. I probably made a blatant error somewhere but I have been stuck for a whole day now. The book gave the answer $0.987862x − 0.155271x^3 ...
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2answers
47 views

Matrix Inversion Test ( Sum of Matrix series)

Friends,I have a set of matrices of dimension $3\times3$ called $A_i$. , Following are the given conditions a) each $A_i$ is non invertible except $A_0$ because their determinant is zero. b) ...
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Accuracy of line intersecting algorithem decrase with large precisions

from the above pic I found the value of x from equation of line p1-p2 and perpendicular line from point a to the Line(p1,p2) .The intersecting point is X ,but the accuracy is less see the result ...
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vector spaces whose algebra of endomorphisms is generated by its idempotents

Let $V$ be a $K$-vector space whose algebra of endomorphisms is generated (as a $K$-algebra) by its idempotents. Is $V$ necessarily finite dimensional? EDIT (Jul 26 '14) A closely related question: ...