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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Inverse of Cartan matrix

The Cartan matrix of the root system $A_n$ looks like, denote it by $A'_n$ $$A'_n= \begin{bmatrix} 2 & -1 & 0 & 0&\ldots & 0 \\[0.3em] -1 & 2 & -1 ...
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8 views

Finding an Orthonormal Basis using Gram Schmidt

Given the set of vectors $S=${${V_1=\binom{1}{4},V_2=\binom{4}{-4} }$} I am to find an orthonormal basis for $R^2$ using the Gram-Schmidt process. I've already worked it out and found the orthonormal ...
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8 views

generalisation of Knonecker matrix product

In the Kronecker matrix product $C = A\otimes B$ we have that $C(i,j)=A(i,j)*B$ where the elements $A(i,j)$ are just numeric scalar values. What if the $A(i,j)$ are matrix operators which act on ...
2
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1answer
15 views

Solution in common for two differential equations

Consider: $E1: y''-4y'+4y=0$ Solution: $y(x)=c_1 e^{2x}+c_2 x e^{2x} $ $E2: y''-2ay'+(a^2-1)y=0$ Solution: $y(x)=c_1 e^{(a+1)x}+c_2 e^{(a-1)x} $ For what values of $a$, $E1$ and $E2$ have ...
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9 views

Matrix Transformations On a Point to Create Fractals

I am working with $3X3$ matrices to perform operations on 2 dimensional geometries, in this a case a 2-D point represented by a $3X1$ matrix. Where the third coordinate is homogeneous. I wish to ...
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2answers
28 views

How would I express the statement “Let H be a subspace of V” in mathematical notation?

How would I express the statement "Let H be a subspace of V" in mathematical notation? Does something like this work? $$ ( \ \ H(\mathbb{R})\subset V(\mathbb{R}) \ ) $$
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12 views

Under what conditions on the field k will all symmetric matrices be diagonalizable?

It's a theorem that if $A$ is an $n \times n$ symmetric matrix ($A = A^{T}$) with real entries, then $A$ is diagonalizable. The proof goes like this: $A$ has a complex eigenvalue, since $\mathbb{C}$ ...
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1answer
41 views

Why is the Det(a)=0 not a subspace? [on hold]

I'm reading my linear algebra textbook, and it says word for word: The following sets is not a subspace when the set of all 2x2 matrices B such that det(B)=0. I just need help trying to understand ...
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7 views

A is a Hermitian projection if and only if it is an orthogonal projection

I need to figure out this property of Hermitian / Orthogonal projections "A is a Hermitian projection if and only if it is an orthogonal projection" Your assistance will be highly appreciated. ...
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2answers
25 views

Help with this easy lemma of linear algebra

I'm trying to demonstrate a theorem of linear algebra and I need to prove this lemma to finish the proof: Let $A=(a_{ij})$ be the matrix representation $T:V\to V$ in the orthonormal basis ...
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1answer
30 views

The Gaussian Integral

Hi I am trying to calculate the expected value of $$ \mathbb{E}\big[x_i x_j...x_N\big]=\int_{-\infty}^\infty x_ix_jx_k...x_N \exp\bigg({-\sum_{i,j=1}^N\frac{1}{2}x^\top_i A_{ij}x_j}-\sum_{i=1}^Nh_i ...
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1answer
27 views

Are these theorems the same?

I'm studying adjoint operators from Schaum's book and I'm confused with these theorems: So the author proves the conjugate transpose $B^*$ is the adjoint of $B$. But some lines after, he states in ...
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1answer
11 views

What is the the projection of vector b onto the matrix A if b is in the Column space of A?

What is the the projection of vector b onto the matrix A if b is in the Column space of A? This is a strange question for me. Can you do a projection in this situation?
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2answers
7 views

Splitting an Indefinite Matrix into 2 definite matrices

I'm attempting to use some quadratic programming techniques to solve a particular optimization problem and my chosen Objective Function is indefinite. I've found some texts online which regard ...
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1answer
22 views

Hilbert space inequality $|\langle x,y\rangle | \leq \epsilon ||x||^2 + C_{\epsilon}||y||^2$

In prelim prep I came across 'given $\epsilon$ there exists $C_{\epsilon}$ such that $|\langle x,y\rangle | \leq \epsilon ||x||^2 + C_{\epsilon}||y||^2$. It is asserted without proof, so I've tried ...
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1answer
24 views

Show that the entries of a matrix are:

For a regression model $y=\beta x$ (note there is no intercept term), show that entries of the matrix $\bf{H} = \bf{X}[\bf{X'}\bf{X}]^{-1}\bf{X'}$ are $h_{ij} = ...
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24 views

Proving that $\mathrm{rank}(P_1+P_2) = \mathrm{rank}(P_1)+\mathrm{rank}(P_2)$

Supposing $P_1$ and $P_2$ two projectors as: $P_1\circ P_2 = P_2\circ P_1$. What is the condition for $P_1+P_2$ to be a projection? If it was the case above then how can I prove that ...
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1answer
20 views

Linearly Independency of vectors [on hold]

Are the vectors $(e^{\frac{\pi}{2}},1)$ and $(110^{\frac{1}{3}}, 1)$ in $\mathbb{R}^2$ linearly independent?
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1answer
12 views

How do I prove the adjoint matrix is the adjoint of the operator

We know that $A^t$ is the adjoint of $A$ if we are in the euclidean spaces: $\langle Au,v\rangle=(Au)^tv=u^tA^tv=\langle u,A^tv\rangle$ (where $\langle u,v\rangle=u^tv$) I couldn't prove the ...
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1answer
22 views

Prove that $sgn(\sigma_1 \circ \sigma_2) = sgn(\sigma_1)sgn(\sigma_2)$

Lete $n\in \mathbb{N}$. Show that the transformation $$sgn: S_n \rightarrow \{\pm 1\}$$ (where $S_n$ is the set of all permutations of the integers in the set $\{1,...,n\}$),given by $\sigma \mapsto ...
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2answers
18 views

Show which matrices are upper triangular orthogonal in $\mathbb R$.

Show which matrices are upper triangular orthogonal in $\mathbb R$. I've tried written the matrix product $Q^T Q$ and I get the following equations: $x_{1,1}^2 = 1$ $x_{2,1}^2 + x_{2,2}^2 = 1$ ...
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1answer
14 views

Show that the image is spanned by the columns of the matrix

No idea how to attempt this, was in an old exam paper for my linear algebra class
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1answer
35 views

These inner products don't match in $\mathbb C^n$

In $\mathbb C^n$, we can define the inner product between $u=\{u_1,\ldots,u_n\}$ and $v=\{v_1,\ldots,v_n\}$ as $\langle u,v\rangle=u_1\overline{v_1}+\ldots+u_n\overline v_n$. I've read in a book that ...
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0answers
17 views

Solution of tridiagonal system

I need a method for solving a system $(x_1 \ldots x_n)$ of type A*x=B where A is a tridiagonal matrix of type \begin{equation}\begin{bmatrix} x & x & \cdots & \cdots\\ x & x & x ...
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1answer
3 views

T-cyclic subspaces and dimensionality?

For each linear operator T on vector space V, find an ordered basis for the T-cyclic subspace generated by vector z. V=R^4 T(a,b,c,d)=(a+b,b-c,a+c,a+d) z=e1 (e1 is the first standard basis in R4 ...
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1answer
20 views

Find two bilinear forms with the same quadratic form over $\mathbb F_2$

Let $V$ be a $K$-vectorspace with a bilinear form $\langle , \rangle$ and the associated quadratic form $q:V \to K, v \mapsto \langle v,v \rangle$. Let $K = \mathbb F_2$. Are there two different ...
2
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1answer
48 views

Eigenvalues of the vectors of

I came across the following problem: "Let $\mathbf a\in\mathbf{R}^n$ be a fixed $n$-component real, non-zero, vector. Let $A^+$ and $A^−$ be real $n\times n$ matrices with components: $$(A^\pm)_{ij} ...
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1answer
43 views

Floating Point Number System

I really have no idea of how to do these questions - in fact I have no idea of how to do any question in the paper - but I have tried to figure out what's going on in the course called Computational ...
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1answer
17 views

Formula needed for calculating probability of recurring events

I'd like to find an answer for calculating the following recurring events: You have X opportunities of picking a ball from a sack. Every time after a ball is picked, the ball is returned to the sack. ...
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1answer
18 views

Generate some random matrix with given rank

Very often for creating new exercises (I teach basic matrix algebra), I need to a find a matrix $A$ such that: it has integer coefficients, not too big (in order to avoid big numbers computations) ...
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4answers
34 views

How to determine the side on which a point lies?

Suppose we have a linear equation and a point in the plane, then how can one determine on which side of the line the point lies?
2
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2answers
52 views

Dual Vector Space embedding

Is there an embedding of any vector space $V$ into $V^*$? As far as I know it is not true. The statement that I know of is that there is natural embedding of $V$ into $V^{**}$ Is there any ...
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2answers
83 views

Why this formula is positive definite?

I have a formula $A(I+GQ)^{-1}(G+GQG)(I+QG)^{-1}A^{\mathrm T}+G$ where $A,Q,G,I\in\mathbb R^{n\times n}$, $A$ nonsingular, $G$ positive semi-definite, $Q$ positive definite, $I$ the identity matrix, ...
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3answers
20 views

Standard equation of a line

I'm a bit confused. I read in many places that the standard equation of a line in $R^2$ is the following: $w_1 x_1 + w_2 x_2 = d$ but I found a resource that mentions it as: $w_1 x_1 + w_2 x_2 + d ...
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1answer
48 views

Linear map $T^2 = -I$. What does it mean?

Assume I have the following linear map: $$T: \mathbb{R}^n -> \mathbb{R}^n, \quad\text{s.t.}\quad T^2 = -I. $$ How can I prove that $n$ is even? Thanks!
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0answers
11 views

When does the leading right eigenvector gives the stationary distribution?

I am trying to make sense of the meaning of the leading right eigenvector in mathematica modeling (applied mathematics). I am interesting in models of the kind $\overrightarrow v(n+1) = M ...
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0answers
24 views

Tridiagonal system [on hold]

I need a method for solving a linear system $(x_1 \ldots x_n)$ of type \begin{equation}\begin{bmatrix} x_1*(k_{11}+k_{12}/(x_1+x_2+ \cdots x_n)) & x_2*k_{13} & x_3*k_{14} & \cdots \\ ...
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3answers
49 views

Prove determinant of $n \times n$ matrix is $(a+(n-1)b)(a-b)^{n-1}$? [duplicate]

Prove $\det(A)$ is $(a+(n-1)b)(a-b)^{n-1}$ where $A$ is $n \times n$ matrix with $a$'s on diagonal and all other elements $b$, off diagonal.
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1answer
21 views

Why is Row/column dimension the number of matrices?

Ok so for example the matrix in REF \begin{bmatrix} 1 & -2 & 5 & 0 & 3 \\ 0 & 1 & 3 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 ...
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1answer
57 views

A question about rational number.

Denote $M$ as a $m\times n$ matrix whose components are all nonnegative integers (actually 0 or 1) and $1$ as the $m$ dimensional vector $(1,1,\cdots,1)$. Show that: There is a vector $x_0$ ...
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reducibility of an operator

This is perhaps a very basic problem in Linear Algebra, which concerns the ability to reduce an operator. An operator $A$ on a finite dimensional vector space $V$ is called reducible if there exists ...
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28 views

sequence of orthogonal polynomial:

Generate the first three members of the sequence of polynomials that are orthogonal with respect to w(x)=x for the discrete inner product over the points xi=(i-1)/4 for 0≤i≤5. so according to the ...
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0answers
15 views

mean number of links in adjacency matrix

I have converted from an individual-level adjacency matrix to one for clusters and I am trying to show mathematically how I programmed up determining the mean number of inter-cluster links. I am not ...
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2answers
19 views

Proving a subset is a subspace

Prove that $ A = \{ \left (t, 2t, 3t \right) :t \in \mathbb R \} $ is a subspace of $\mathbb R^3$. $Proof$: If $t = 0$, then $(t, 2t, 3t) = (0, 0, 0)$. So, $(0, 0, 0) \in A$. Suppose $(t, 2t, ...
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0answers
24 views

Determinant, Rank

Lat $K$ be a field, $K\subset \Bbb C$. $a_0,a_1,a_2,\dotsc$ is a sequence, $a_i\in K, i=0,1,2,\dotsc$ For integers $s,m\geq0$, Defined $$A_{s,m}=\begin{bmatrix} a_s & a_{s+1} & \dotsc ...
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14 views

How to deal with x* when solving complex-variable linear equation(s) of x?

The theory of linear algebra can be directly applied to linear equation(s) of complex variables with the form \begin{equation} \sum_i a_i x_i=c\ldots\ldots(1) \end{equation} with $a_i,c\in ...
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17 views

Determine if a sub-set of n points exactly solves an over-determined linear, homogeneous system of equations

Given a $m \times n$ matrix $\mathbf{A}$ that describes the homogeneous system of linear equations $\mathbf{A} \mathbf{x} = 0$, I want to find the the largest number $s$ of individual points (rows) in ...
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0answers
11 views

What is meant by a Rank 1 Update?

What is meant by a rank-1 update in linear algebra? I've performed the calculations but I don't understand why it is called a rank-1 update.
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22 views

Finding mathematical relation of matrices with reverse indices

I am designing a simple game, I have faced this problem to get the mathematical relation between two kind of tables: MATRIX A MATRIX B As you can see the table A (or Matrix A) is the normal ...
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17 views

linear algebra - fourier coefficients of piecewise

Find fourier coefficients of given function: f(t) = {-1 if t $\leq$ 0; 1 if t > 0} so do I do this? $a_{0} = \int_{a+-\pi}^{a+\pi}1$, $a_{k} = \int_{a+-\pi}^{a+\pi}1*cos(kx)$, $b_{k} = ...