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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Operatornorm of $(\mathbb{R}^d, \|\cdot\|_1) \to (\mathbb{R}^d, \|\cdot\|_{\infty})$

Determine the operatornorm of the mapping $I:(\mathbb{R}^d, \|\cdot\|_1) \to (\mathbb{R}^d, \|\cdot\|_{\infty})$! Unfortunately I haven't many ideas for this task. I know that the definition of the ...
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6 views

Construct piecewise formula for position s, given accel position, velocity

I'm writing an algorithm to control the position of a motorised system, and I'm trying to construct a formula which I can then translate into C. I'd like some examples of piecewise formulas which ...
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3answers
42 views

Baker-Campbell-Hausdorff/Zassenhaus formula to first order in one matrix

Is there a closed-form expression for the term of $e^{t(c \hat{X} + d \hat{Y})}$ that is first-order in $d$, where $t$, $c$, and $d$ are scalars and $\hat{X}$ and $\hat{Y}$ are finite-dimensional ...
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1answer
44 views

Implementing gradient descent based on formula

The gradient descent algorithm is given as : repeat { $$\displaystyle \theta_j := \theta_j - \frac{1}{m} \alpha \sum_{i=1}^m (h_\theta(x^{(i)}) - y^{(i)}) x^{(i)}_j $$ } Given these values : <...
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Whether a given algebra is the algebra of endomorphisms for a vector space.

Let $\mathbb{F}$ be a field and let $A$ be an associative unital $\mathbb{F}$-algebra. Is there a criterion to let me know if $A$ is isomorphic to the algebra $\mbox{End}(\mathbf{V})$ of endomorphisms ...
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10 views

How do I compute the area of this parallelogram

Given vectors $a,b$ and the ribs of parallelogram are $2a +3b = A$, $a-2b = B$. Also given $a \times b = (-1,2,2)$. Compute the surface of the parallelogram. I'm not sure where I saw but I think it ...
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13 views

Showing properties of a function and its inverse image

I tried proving the following question but did not get too far. Let $\ f:A \to B$ be a function and $\ f^{-1}(Y)$ be the inverse image of $\ Y\subseteq B$ on $\ f$. Consider the following ...
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12 views

commutativity of log(I + A) and log( A−1) (matrix function)

I'm self-(re)learning linear algebra since the beginning of the summer, and i have a problem with the following exercice entitled additive logarithmic. If i'm right, we need to prove the ...
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65 views

Is this an Example of a Dual Space? [on hold]

Is the set of possible bases that I describe $∀(e_1,e_2,e_3)$justSlash$∀(e_1,e_2,F(e_1, e_2))$ F defined V, \times. v=e_1 \times e_2*for any linear vector space of dimension 3* and their linear ...
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25 views

Functional equation of $f(n)=\sum_{k=0}^{n-1}g\left(x+\frac{k\pi}{n}\right)$

Suppose the function $f(n)$ is given by: $$f(n)=\sum_{k=0}^{n-1}g\left(x+\frac{k\pi}{n}\right)$$ Where $x\in\mathbb{R}$. I am looking for a formula that enables me to express $f(n)$ as : $$f(n)=\sum ...
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1answer
392 views

Finding a matrix representation for two Grassmann numbers.

This question is more general in the sense that I want to know how one finds a particular (say matrix) representation for any object. For the case of Grassmann numbers we have from Wikipedia the ...
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1answer
713 views

Column and Row Picture for Singular System of 100 Equations (Strang P55, 2.2.32)

Start with 100 equations $\color{#8F00FF}{A}\mathbf{x} = \mathbf{0}$ for $\mathbf{x} = (x_1, ..., x_{1oo})$. Suppose elimination reduces the 100th equation to $0 = 0$, so the system is "singular". ...
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15 views

SVD with degenerate singular values

I'm using SVD to do some kind of low rank approximation, basically I have to compute the largest eigenvalues, I also tried different routines, ...
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70 views

Questions on color theory, expressed in linear algebra

I'm reading into color theory and there were a few questions which I asked myself along the way, maybe you can put me forward to some source where I can find answers or give them directly. The ...
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1answer
33 views

Is the finite dimension of a vector space over the complex numbers half the dimension of the same vector space considered over the reals?

Consider a vector space with basis ${b_{1},..,b_{n}}$ and complex scalars. This obviously has dimension n. Now consider the same exact set of vectors, except with real scalars. I am thinking the ...
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1answer
18 views

How to find the total derivative of a function $f_a(y(t),x(t))$ subjected to parametric change with the parameter $a$

It is well known to find the total derivative of a function $f(x(t),y(t))$. I consider it as $Td_f$. What, if the function depends upon some parameter, say, $a$. Then, how to find the total derivative ...
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1answer
52 views

Finding non-trivial solutions for the system of linear algebraic equations

Suppose we have a system of $n$ linear algebraic equations where $n>1$ is a positive odd integer. The matrix $A=\{a_{ij}\}_{i,j=1}^n$ of this system has the following properties: $a_{ii}=0$ for ...
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1answer
42 views

Find $B(B^{T}B)^{-1}B^{T}$.

To find: $$B(B^{T}B)^{-1}B^{T}$$ for $B=[0,1,-1]^T$ I have $$\begin{bmatrix} 0\\ 1\\ -1 \end{bmatrix} \left ([0,1,-1]\begin{bmatrix} 0\\ 1\\ -1 \end{bmatrix} \right )^{-1}[0,1,-1]$$ but ...
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Find a matrix with determinant equals to $\det{(A)}\det{(D)}-\det{(B)}\det{(C)}$

Assume I have 4 matrices $A,B,C,D\in\Bbb{R}^{n\times n}$. I want to build a matrix $E\in\Bbb{R}^{m\times m}$ such that: $$\det{(E)}=\det{(A)}\det{(D)}-\det{(B)}\det{(C)}$$ under the following ...
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1answer
44 views

Is a vector space isomorphic to the kernel $\oplus$ image of a map out of it?

Let $f:V\to W$ be a linear map of finite-dimensional vector spaces. By simply counting dimensions and using rank-nullity, it is clear that $V\cong \mathrm{im}\,f\oplus\mathrm{ker}\,f$. I want to know ...
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Preposition about the Entries of the Product of Markov Matrices.

Definition: A Markov matrix is an $n \times n$ complex matrix with the sum of the elements in every column equal to 1. My task is to prove that: If A, B are Markov matrices such that $|a_{ij}|\leq1$ ...
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1answer
31 views

Row sum of inverse of a matrix

Let's say I have a matrix A, $$A= \begin{bmatrix} a_{11}& a_{12} & a_{13} \\ a_{21}& a_{22} & a_{23} \\ a_{31}& a_{32} & a_{33} \end{bmatrix} $$ All the elements of A are ...
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2answers
16 views

Finding an approximate function using orthonormal basis

I'm trying to take a function in $C_0[0,1]$ space (let's call this $f(x)$) and trying to find the best approximate of $f(x)$ at $P_2[0,1]$ space (let's call this approximate $p(x)$). Note that $P_2[0,...
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0answers
27 views

factors sum to 1

If I have factors of linear operators say $$(a_1 + A)(a_2 + A)(a_3 + A)\cdots(a_n + A) = 0$$ $A$ being an linear operator(i guess it really doesn't matter its operator or not) why $$\sum_{n} \frac{...
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3answers
5k views

Examples for proof of geometric vs. algebraic multiplicity

Here you see a supposedly easy proof of a well-known theorem in linear algebra: Although I know I should understand this, I don't :-( Obviously there are too many indices and stuff, so I don't see ...
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1answer
22 views

Least squares solutions and orthogonal projection?

I found the least squares solution for the following inconsistent system of equations: $ x_1 - x_2 = 0$ $ x_1 + x_2 = 5 $ $-x_1 + x_2 = 2$ , which turned out to be $ \begin{bmatrix} 2\\ 3\\ \end{...
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1answer
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Write summation of vector outer products into matrix form

My question is as follows: Given the weighted summation of vector outer products $\sum_i\sum_jh_{ij}{\bf v_i}{\bf u_j}^T$, where $h_{ij}$ is the weight, and ${\bf v_i,u_j}$ are column vectors, I was ...
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1answer
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Conjugacy of Cartan subalgebras

This is probabably a very silly question, stemming from some fundamental misunderstanding I have of the relevant definitions, but I am stumped by it. I know that any two Cartan subalgebras of $\...
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2k views

Determine point of interesction of plane with axis given points of plane

Q: The points $(2,-1,-2)$, $(1,3,12)$ and $(4,2,3)$ lie on a unique plane. Where does the plane cross the z-axis. I understand that the point of intersection would occur at $(0,0,z)$ and I have to ...
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Don't know how to enter this into webwork [on hold]

I know the vectors are (-3-i ; 2) and (-3+i ; 2) however no matter which way I enter it into the program, it regards my answer as incorrect. How am I to enter the answer?
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2answers
155 views

What is the nullity of an onto transformation?

For a $5 \times 13$ matrix, with $T(x) = Ax$, what is the nullity of $A$ if $T$ is onto? I can't figure out what it would be...
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33 views

Smallest possible value of the norm?

The vectors $ \vec{u_1} = \begin{bmatrix} 1 \\ 1 \\ 1\\ 1 \end{bmatrix} $ and $ \vec{u_2} = \begin{bmatrix} 1 \\ -1 \\ 1\\ -1 \end{bmatrix} $ are orthonormal in $ \mathbb{R}^4$....
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2answers
53 views

Gradient and Hessian of function on matrix domain

Let $A \in R^{k \times p}$. Define $f(X) : R^{p \times k} \rightarrow R$ to be $f(X) = \log \det(XA + I_{p})$, where $I_{p}$ is a $p \times p$ identity matrix. I want to know what is the gradient and ...
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Completing a semi-vector space to a vector space

If we have a semi-vector space $U$ (as defined here), what do we have to additionally demand from $U$ such that we can complete it to a vector space $\tilde{U}$ via $U\xrightarrow{\iota}\tilde{U}$ and ...
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1answer
51 views

How do I prove this statement in linear algebra?

I had a test about a week ago and I want to know the answer to this question: Given an orthonormal basis $B$ spanned by $\{ v_1, v_2 , v_3 \}$ of ${\mathbb R}^3$ Prove that for every $v \in {\mathbb ...
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1answer
36 views

Trying to visualize and understand double dual space

Currently I am reading "Finite-dimensional vector spaces" by Paul Halmos. I would have a question regarding the theorem on page 25. It says: If $V$ is a finite-dimensional vector space, then ...
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Find non-diagonal matrices $A$ and $B$ such that $B^TAB$ is diagonal

Here $B^T$ denotes the transpose of $B$. $A$ and $B$ are invertible $3\times 3$ matrices with integer entries. $A$ is symmetric positive definite with at most two zero entries. We want the ...
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1answer
25 views

Solving a recurrence relation of conditional probability functions

Suppose you have the recurrence relation for a probability function Q: $$Q(n_1,n_2|n) = Q(n_1-1,n_2|n-1)\frac{n_1-1}{n-1} + Q(n_1,n_2-1|n-1)\frac{n_2-1}{n-1}$$ where $n = n_1 + n_2$ and the ...
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Application of Structure Theorem to Prove Simultaneous Diagonalizability and Group of Units of Cyclic Groups

I am reading these notes on Modules over PID. Exercise 67 (pg 24) asks to prove that: Problem. Let $A$ and $B$ be $n\times n$ matrices with complex entries. Then $A$ and $B$ are simultaneously ...
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Reflection relating two subspaces

Let $S_1, S_2 \subseteq \mathbb{R}^n$ be two linear $k$-dimensional subspaces. Does there always exist a hyperplane $H$ such that $S_1 = R_H S_2$, where $R_H$ denotes the orthogonal reflection across $...
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72 views

An inequality on the rank of a block matrix

Let $\mathbb F$ be a field, and let $r_1, r_2, s_1, s_2$ be positive integers. Consider the matrix $$X:=\left[\begin{array}{cc} A & B \\ C & D \end{array} \right],$$ where $A \in \mathbb F^...
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2answers
446 views

Maximum and minimum of determinant of matrices with entries from $\{0,1\}$ or $\{-1,0,1\}$

Maximal and Minimal value of $\bf{3^{rd}}$ order determinant whose elements are from the set $\bf{\{0,1\}}$. Maximal and Minimal value of $\bf{3^{rd}}$ order determinant whose elements are from the ...
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1answer
317 views

$\dim C(AB)=\dim C(B)-\dim(\operatorname{Null}(A)\cap C(B))$

Let $A \in M_{n \times m}\left(F\right)$ and $B\in M_{m \times p}\left(F\right)$ for a field $F$. Prove: $\dim C(AB)=\dim C(B)-\dim(\operatorname{Null}(A)\cap C(B))$, where $C(X)$ denotes the column ...
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34 views

Eigenvalue perturbation of singular matrix

Consider a Hermitian matrix $\mathbf{A_0} \in \mathbb{C}^{N \times N}$ with one singularity, i.e. its eigenvalues in increasing order are: \begin{equation} 0 < \lambda_2 \leq \lambda_3 \leq \cdots \...
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12 views

Calculating the null space and the column space

Suppose I have matrices $A_{n\times n},B_{n\times n}$ and the appended matrix $[A \hspace{0.25cm} B]_{n\times 2n}$, suppose both $A,B$ are of rank $n-1$, could anyone tell me how the followings are ...
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1answer
55 views

Why is this matrix symmetric?

There is an example in the Convex Optimization lecture notes, Boyd. He just said in the lecture that the matrix which is underlined in red color is symmetric! How can we claim that when there is no ...
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0answers
15 views

Generalise Expression to slice Circulant Matrix

Suppose I have $4 \times 4 $ circulant matrix , $$A=A(0:3,0:3)=A(:,:)=\begin{bmatrix} 1 & 2 & 3 & 4 \\ 4 & 1 & 2 & 3 \\ 3 & 4 & 1 & 2 \\ 2 & 3 & 4 & 1 \...
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1answer
18 views

Is the condition number of unitary matrix always equal to 1?

I know that the 2-norm condition number $\kappa (\textbf U)={||\textbf U||_2}{||\textbf U^{-1}||_2}$ of a unitary matrix $\textbf U$ is always equal to 1. Is this true for all induced matrix norms, i....
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19 views

Entanglement of 3-qubit states

Given a separable 3-qubit state φ = φ0 ⊗ φ1 ⊗ φ2 with φi= ai0|0> + ai1|1>, |0>, |1> being the computational base. φ thus can be written as φ = b000|000> + b001|001> + b010|010> + b011|011> + ...
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0answers
15 views

Calculating the coefficients of a separable 2-qubit state

Given a separable 2-qubit state φ = φ0 ⊗ φ1 with φi= ai0|0> + ai1|1> φ thus can be written as φ = b00|00> + b01|01> + b10|10> + b11|11> with bij = a0ia1j. Now let some bij be given, i.e....