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Questions tagged [linear-algebra]

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 concerning matrices, use the (matrices) tag. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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1answer
28 views

Set of real valued functions defined and continuous on closed interval $[0,1]$ [on hold]

Prove that $C[0,1]$ such that $f(3/4)=0$ is not a vector space?
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1answer
24 views

Find the eigenvalues of a polynomial transformation

Say $$F_2[X]:=\{p(x)=ax^2+bx+c:a,b,c \in F\}$$ And the linear operator T:$F_2[X] \to F_2[X]$ defined as: $$T(ax^2+bx+c)=(2a+6b+5c)-(8a+b)x+(c-2a)x^2$$ How can I find the eigenvalues and ...
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2answers
14 views

Geometric proof for Composition bound property of operator norms?

This is just a curiosity. For linear transformations $A$ and $B$, $||AB|| \le ||A|| \cdot ||B||$ where$||\cdot||$ denotes the operator norm (Of course provided $AB$ exists.) This fact has a proof, but ...
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1answer
28 views

Solving weighted least squares with non-negative constraints

I have the optimization problem $$ \begin{align} \min_{\mathbf{P} \geq 0} \|\mathbf{A\odot(X-PQ^\top)}\|^2 + \frac{\|\mathbf{P}\|^2}{2} \end{align} $$ $\odot$ is the Hadamard product, $\mathbf{A,X,P,...
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0answers
27 views

Are there degrees or categories of orthogonality?

The vectors $\begin{bmatrix}a \\ 0 \\ 0\end{bmatrix}$, $\begin{bmatrix}0 \\ b \\ 0\end{bmatrix}$, and $\begin{bmatrix}0 \\ 0 \\ c\end{bmatrix}$ are orthogonal under the dot product. The vectors $1$, $...
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0answers
33 views

Dimension and Base of $(E_{12} + E_{21} + E_{23} + E_{32})A = 0$ Matrix.

You have a set: $W = {A ∈ Mn(R) | (E_{12} + E_{21} + E_{23} + E_{32})A = 0}$. You need to find out if this set is subspace of $M_n(\mathbb{R})$. I proofed it that way: $(E_{12} + E_{21} + E_{23} + ...
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3answers
50 views

Find a matrix 4x4 that has no eigenvalues on $R^4$

I had to find a matrix 2x2 and 4x4 that have no eigenvalues, for the 2x2 it was not that hard to do $a_{11}= 0$ $a_{12}= 1$ $a_{21}= -1$ $a_{22}= 0$ so that the possible eigenvalues are $det(xId-A)=x^...
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0answers
9 views

Reformulating a high-rank linear system into a block-matrix equation

I have on my hands a linear system of equations of the following form $$ \sum_{j=1}^K\sum_{q=1}^N A_{ijpq} x_{jq} = b_{ip} \quad(i=1\dots K,p=1\dots N) $$ in which the $x_{jq}$ are unknown and the ...
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0answers
15 views

If I have a matrix in the form $Ax=B$, why must I have $\begin{bmatrix}i\\j\end{bmatrix}=β\begin{bmatrix}-2\\1\end{bmatrix}?$

I have inserted numbers in place of i and j, and have successfully deduced the value of β as a coefficient. However, i am unsure why we must have that equation if we know $A\begin{bmatrix}i\\j\end{...
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1answer
36 views

matrix standardization, inferring technique

I am reading a research paper in which authors perform a matrix standardization but do not explain the actual procedure. The original 3x3 matrix is: ...
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1answer
10 views

For $V = \sum_{s=1}^{t} A_s A_{s}^T$ to be non-singular $(A_s)_{s=1}^{t}$ needs to span $R^d$

I am reading a book on bandits algorithm and inside a proof it says the following: Let $(A_s)_{s=1}^{t}$ be sequence of vectors in $R^d$. Construct a matrix $V$ such that: $$ V = \sum_{s=1}^{t} A_s ...
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0answers
6 views

Spectral methods on semiring matrices

It is always very useful to have the spectral decomposition of a matrix or to know something interesting about eigenvalues and eigenvectors of a matrix of interest. In algebraic algorithms, one often ...
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1answer
21 views

Kernel of orthogonal projection on an eigenspace

let $Q$ be a $d\times d $-matrix and $P:\mathbb{R}^d \to \mathbb{R}^d$ be the orthogonal projection on the eigenspace $E_0 $ of $Q$. Why is the kernel of the projection the sum of the other ...
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0answers
12 views

scalar triple product for continuous functions

How is the scalar triple product for continuous functions, rather than vectors? Can it also be written using the integral of the product of three continuous functions?
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1answer
26 views

Algebraic problem

Am not sure about the title for this problem. Anyone can edit otherwise In a test,a professor awards 5 marks for every question a student gets right and deducts 2 marks for a wrong answer.In that ...
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2answers
13 views

The conjugate of a scalar is the same scalar in a matrix-scalar multiplication?

I am reading the book, Applied Linear Algebra and Matrix Analysis. When I was doing the exercise of Section 2.4, Page 97, I thought the equation is right as followed: $(cA)^{*} = cA^{*}$ But answer ...
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0answers
46 views

How many bases can be formed given a set of vectors? [duplicate]

Let $v_1,\cdots,v_n$ be a set of vectors such that the generated vector space has dimension $r$. How many different bases with dimension $r$ can be formed from these vectors? Don't understand the ...
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0answers
34 views

Help with Limit 9

I have the following system of partial derivatives: $$\frac{\partial Y}{\partial K}=\frac{1}{K}\left ( Y-\frac{\partial Y}{\partial L}L \right )$$ $$\frac{\partial Y}{\partial L}=\alpha \left (\frac{...
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3answers
36 views

Easier way to solve a LES (only pen/paper, no calculator)

I want to solve the following Linear Equation System with only pen and paper; $$ 470 = x_A - \frac{3}{10}x_B \tag{1} $$ $$ 940 = x_B - \frac{2}{10}x_A \tag{2} $$ I attempt to solve for $x_A$ by ...
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0answers
73 views

Matrix inverse algorithm that works for any unitary ring

Is there any algorithm to find out if a given square matrix has an inverse (which is both left and right inverse), and compute the inverse, if there is one for any unitary ring, without assuming ...
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1answer
42 views

Questions in Linear Algebra

Let $A \in\mathbb(C)^{p\times n}, C \in \mathbb(C)^{p\times q}$. Show that you cannot have more than one matrix $B\in\mathbb(C)^{n\times q}$ s.t. both of the properties hold: 1) $AB = C$ and 2) $B^{...
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3answers
669 views

Burning a rope to count time

A rope burns irregularly in 16 minutes and costs 32 rupees, while a second rope burns also irregularly in 7 minutes and costs 14 rupees. Both can be lit only at one end and can be turned off and lit ...
2
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1answer
28 views

Show that $(A^o)^o = A$ if A is compact and convex.

I am working on exercises for a convex analysis course and I am slightly stuck on the following; Suppose that $A \subset \mathbb{R}^n$ is closed, bounded (compact) and conex. Define the set $$ A^o = ...
2
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1answer
35 views

After multiplying a positive definite matrix several times to 'a vector A', still less than 90 degree between the 'vector A' and the 'mapped vector'?

My question Would the $\theta$ be still less than 90 degrees in vT * Mk v = ||v|| * ||Mk v|| * cos $\theta$, if the matrix M is positive definite? Background Information Let's suppose that v (...
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1answer
21 views

Show that the function $||u|| = \sqrt{2|u_1|^2+5|u_2|^2}$ is a norm on $V$.

I'm given: Let $V$ be the real vector space $\mathbb{R}^2$, and $u = [u_1 \, u_2]^T \in V$. Show that the function $||u|| = \sqrt{2|u_1|^2+5|u_2|^2}$ is a norm on $V$. Then, determine whether this ...
0
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1answer
19 views

Prove that $S = \{A \in GL_n(K) | AJA^t = J\}$ is a subgroup of $GL_n(K)$

Given a field $K$, $n \in \mathbb{N}$ and $J \in K^{n \times n}$, show that: $$ S = \{A \in GL_n(K) | AJA^t = J\} $$ is a subgroup of $GL_n(K)$. To show that S is a subgroup, we must show: It must ...
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1answer
13 views

Proof on characteristic values

If $V$ is a vectorial space of finite dimension over a field $F$ and T is a invertible linear operator over the field $V$, also, say that $\lambda \in F^*$ Prove that $\lambda$ is a ...
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0answers
20 views

How to obtain the Q matrix from QR decomposition using Fast Givens rotation [on hold]

I am not able to obtain Q matrix correctly when I attempt QR decomposition using fast Givens rotation. Would somebody kindly help along with a numerical example?
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0answers
32 views

The span of a finite number of vectors in a normed vector space is closed

I want to prove that the span $S$ of a finite number of vectors $v$ in an arbitrary normed vector space $V$ is a closed set. My plan is to show that all convergent sequences ${x_n}$ contained in the ...
0
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1answer
27 views

Find the image using the matrix relative to B and B'

I understand how to find the image of a transformation using the standard matrix, however, I don't quite get how you would use two bases to obtain the image. I've searched my whole Linear Algebra ...
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1answer
26 views

Linear transformation with change of ordered basis

Let the linear transformation $ D: P_3 \rightarrow P_2 $ be defined as $ D(f(x)) = f'(x) $. $ B = \lbrace 1, x, x^{2}, x^{3} \rbrace, \\ C = \lbrace 2+x+x^{2}, -2-x^{2}, 1-x \rbrace $ Find the ...
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0answers
27 views

Find a basis for the kernel and image of matrix A

I need to find a basis for the kernel and image of the matrix A $A = \left( \begin{array}{ccc} -12 & 6 \\ 4 & -2\\ -8 & 4 \end{array} \right)$ But I am unsure how to do that. For the ...
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1answer
51 views

Is the axiom of additive inverse necessary for the definition of vector space?

Is there an example that satisfies the other five axioms but is not a vector space? The usual definition of vector space requires $(V, +)$ to be an Abelian group. If we relax this condition so that $(...
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0answers
19 views

finding L in LPU decomposition

I 'm having a trouble finding the inverse of L, lower triangular matrix, when given the original matrix and U, upper triangular matrix, in LPU form ...
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0answers
26 views

Finding a basis for the physical constants?

I was not able to find much on this topic besides this article, and don't know if this is even something worth considering, but it has been stuck with me for a while. Since any given set of data is ...
0
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1answer
41 views

Finite sum of reciprocal factorials

Does someone know how to calculate sum $\sum_{k=1}^n \frac{1}{(n-k)!\cdot k} $? I was working something with matrices, i.e. calculating number of all cycles in Coates digraph, and got this weird ...
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0answers
12 views

How to show that the operator on the space of functions with periodic boundary conditions is symmetric

I have the steady state boundary value problem with periodic boundary conditions $u-u''=f, u(-1)=u(1), u'(-1) =u'(1)$. I need to show that the operator $L_p: C_p^2[ -1,1]\to C[-1,1], L_pu=u-u''$, ...
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1answer
33 views

Is it possible to determine the constituents of a matrix product, given the result?

Suppose we have a set $M$ of two or more matrices such that every matrix product $X$ composed of matrices drawn with replacement from $M$ is unique. Is there a set $M$ for which we can determine the ...
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0answers
32 views

Why is $ \max_{i} | \lambda_i(A) | \leq \| A \|_P $?

I was told: $$ \max_{i} | \lambda(A) | \leq \| A \|_P $$ I tried thinking through it. So the operator norm is defined as: $$ \| A \|_P = \sup_{y \neq 0} \frac{ \| A y \|_P }{ \| y\|_P } = \sup_{ \| ...
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1answer
13 views

Proving infeasibility using Duality

suppose we have the linear program min{$c^Tx: Ax \leq 0, x \leq 0$} and its corresponding dual max{$0^Tx: A^Ty \geq 0, y \leq 0$}. How can we show that the Dual is infeasible? I started by ...
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2answers
53 views

Solving Linear Algebra Equation for Matrix Power

I'm stuck trying to solve a matrix equation for a power of a matrix. If $\boldsymbol{A}$ is a 2x2 matrix, and $\boldsymbol{x}$ is a 2 x 1 vector, I have the equation: $\boldsymbol{A}^s \boldsymbol{x} ...
2
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1answer
44 views

A convex combination of unitary transforms converts any matrix to identity

Question Show that there exists a set of unitary matrices $\{U_i\}$, and probability $\{p_i\}$, such that for any $n \times n$ matrix $A$ \begin{equation} \tag{1} \sum_{i} p_i U_i A U^{\dagger}_i = \...
1
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1answer
57 views

Automorphism of a matrices ring

Let $R$ be the ring of $3 \times 3$ matrices with coefficients in $\Bbb Z_5$. For every $g \in GL_3(\Bbb Z_5)$ prove that the function $$f\colon R \rightarrow R$$ defined as $$x \mapsto g^{-1}xg$$ ...
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0answers
13 views

Change of basis (vector space $\mathbb{R}_{2n}[X]$)

I have a problem with this exercise : Let $n \in \mathbb{N}^{\ast}$ and $a, b \in \mathbb{R}$ ($a \neq b$). Show that $\{(X - a)^{k}\}_{0 \leq k \leq 2n}$ is a basis of $\mathbb{R}_{2n}[X]$ (for that ...
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0answers
56 views

Is there a name for a matrix with symmetric but inverse entries?

A matrix $A$ is symmetric if it is equal to its transpose. Then, the following equality holds between the entries of this matrix: $$a_{ij}=a_{ji}$$ Is there an established name for a similar matrix in ...
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1answer
12 views

A question for clarification with respect to a proof in an article on linear transformations on nonnegative matrices

I am reading an article by H. Minc: "Linear transformations on nonnegative matrices" (source https://core.ac.uk/download/pdf/82808273.pdf). There is a step in the proof of theorem 2 which I do not ...
0
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1answer
51 views

A statement about reduced row echelon form

According to Nicholson's linear algebra : The matrix $R$ has $r$ leading ones (since rank $A = r$) so, as $R$ is reduced, the $n \times m$ matrix $R^T$ contains each row of $I_r$ in the first $r$ ...
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0answers
24 views

Ways to approximate functions using polynomials

Other than Taylor's theorem what other methods can be used to approximate a function that ultimately reduces it down to a polynomial?
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1answer
51 views

How would be a formal answer for an automata geometry problem?

Let an automaton $A$ sit on point $O$ $(0,0)$ and turned to the North. That automaton can execute only any combination of three different commands in each step: Move one unit forward Turn 90 degrees ...