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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Solutions of a linear system writen as linear combinations of vectors

I have a linear system of 2 equations and 5 variables. The answer in the book is a linear combination of 3 vectors, each one being multiplied by a different parameter. I solved it by adding the first ...
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about derivative of a matrix and trace

I have checked it up the following derivation of a formula:" The question that I have is why the author uses the trace in the third part; supposedly it uses a formula derived from the properties of ...
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Suppose $\langle\cdot,\cdot\rangle_1$ and $\langle\cdot,\cdot\rangle_2$ are inner products on $V$ such that…

Suppose $\langle\cdot,\cdot\rangle_1$ and $\langle\cdot,\cdot\rangle_2$ are inner products on $V$ such that $\langle v,w\rangle_1=0$ if and only if $\langle v,w\rangle_2=0$. Prove that there is a ...
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Prove that there is an inner product on $\mathbb{R}^2$ such that the associated norm is given by…

Suppose $p>0$. Prove that there is an inner product on $\mathbb{R}^2$ such that the associated norm is given by $||(x,y)||=(x^p+y^p)^\frac{1}{p}$ for all $(x,y)\in\mathbb{R}^2$ if and only if ...
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Schauder basis and Eigenbases

There are several question in this site comparing different basis functions including Schauder basis and others, but I could not connect the difference between the Schauder basis and Eigenbasis ...
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8 views

A 2D smoothing convolution filter

I'm trying to find the right form of a 2D filter that will do the following to a matrix after linear convolution: Let A = [ ? ? ?] [ ? ? ?] [ ? ? ?] and B = ...
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Inverse properties of l_1 normed matrices

Let's take the adjacence matrix $A$ of a graph $G$. We call $\bar{A}$ the row $L_1$ normalized matrix obtained from $A$. Let's take some $\alpha \epsilon [0,1)$. $(I-\alpha\bar{A})$ is strongly ...
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Understanding matrix property

I am reading about matrix property from here. On page 2 of pdf (equation 2.2), it says if $A$ is a matrix and $U$ a row-echelon form of $A$ then $$|A| = (-1)^r \alpha |U| ...
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WLOG doubt: why can we assume that two disjoint linear subspaces of dimension $n$ in $\mathbb{P}^{2n+1}$ are given by the following equations…

Let $H_1,H_2$ be two linear disjoint subspaces of dimension $n$ in $\mathbb{P}^{2n+1}$. Let $(x_0:\cdots:x_n:y_0:\cdots:y_n)$ be homogeneous coordinates in $\mathbb{P}^{2n+1}$. My question is: ...
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explicitly splitting the hamilton quaternions over local fields

For simplicity, lets first consider the hamilton quaternions $$ H = \left(\frac{-1,-1}{\mathbb{Q}}\right)$$ This is the central division algebra over $\mathbb{Q}$ with $\mathbb{Q}$-basis given by ...
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Sum of orthogonal matrices

Consider the subspace $\mathbb{R}^m$ with usual inner product.Let $S_1$ and $S_2$ subspaces of $R^m$, $P_1\in M_m(\mathbb{R})$ the orthogonal projection matrix on $S_1$ and $P_2\in M_m(\mathbb{R})$ ...
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Linear Algebra - Straight line determined by two distinct points

Let A and B be two distinct points em $R^3$. Prove the straight line r(A,v): $P = A + v*t$, $t \in R$, where $v = B-A$, is the only straight line which contains A and B.
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Unknown functions yield a given determinant

I am trying to develop a nomogram which simultaneously shows the exact Fisher equation $(1+u) = (1+v)(1+w)$ and its linear approximation $u \approx v + w$. This amounts to finding twelve smooth ...
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Proof of linear independence of non-empty subsets

The question states: Show that if $S = \{v_1, v_2, \ldots , v_r\}$ is a linearly independent set of vectors, then so is every non-empty subset of $S$. I understand that if $r>n$, $S$ is ...
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norm of matrix 1 [on hold]

SHOW ∥A∥1=∥AT∥∞? i dont solve.....
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How can we memorize the formula for the determinant of a $4\times4$ matrix?

This is the formula for the determinant of a $4\times4$ matrix. . 0,0 | 1,0 | 2,0 | 3,0 0,1 | 1,1 | 2,1 | 3,1 0,2 | 1,2 | 2,2 | 3,2 0,3 | 1,3 | 2,3 | 3,3 . ...
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How to count algebraic multiplicities to show $\nexists$ an eigenbasis for $A$?

If $A=\begin{bmatrix}1&1&0\\0&1&1\\0&0&1\end{bmatrix},f_A(\lambda)=(1-\lambda)^3 \,\text{and } E_1=\text{ker ...
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For which values of lambda do the following vectors form a linearly independent set in R^3

I've seen this same question, but asking for linearly dependent, not linearly independent. $$ V_1= \left(a,\, \frac{-1}{2}, \,\frac{-1}{2}\right),\;\; V_2= \left(\frac{-1}{2},\, a, ...
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Solution space of Linear homogeneous differential equation

The solution space of a L.H.D.E of order n is a vector space spanned by n base vectors, right? So any solution is then a vector of the solution space -> a linear combination of the base vectors. But ...
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Fill in the missing entries of matrix $Q$ to make it orthogonal

I am given the following matrix $Q$: $Q=$ where $p1,p2,...,p8$ are unknowns. I need to make $Q$ into an orthogonal matrix. It occurs to me that $v1 =\{1,1,1,1\}$ and $v2=\{2,1,0,-3\}$, but I'm ...
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What does affine invariance mean in the context of the Newton's method?

The textbook Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (by Ascher, Mattheij, and Russell) states on page 329: [W]e observe that Newton's method is affine ...
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Affine Transformations: Book to Study over the Summer

I've briefly heard of affine transformations in both linear algebra and calculus and I'd like to find a good book on the subject to study over the summer. So what's a good undergrad-level book on ...
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Orthogonality and inner product

Let $A\in M_2(\mathbb{R})$ a positive definite matrix and the application $F:\mathbb{R}^2 \times \mathbb{R}^2\rightarrow \mathbb{R}$ $$F(x,y)=y^tAx$$ If ...
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Nilpotent matrix similar to a matrix $[0,X]$ where $X$ is full column rank.

I am trying to prove that a nilpotent matrix $N$, which has a Jordan Form consisting only of blocks which are order 2 or greater, is always similar to a matrix $\begin{bmatrix}0 & X\end{bmatrix}$ ...
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20 views

Showing linear functionals are linearly independent

In general: Given $f_1,f_2,...,f_n\in W^*$. To show they are linearly independent, will it be enough to take the standart base of $W$,$B=\{e_1,...,e_n\}$ and its dual base $B^*=\{g_1,g_2,...,g_n\}$ ...
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Where does $\Lambda=P^{-1}AP$ come from?

How do we derive the fact that if a matrix is diagonalizable then we can diagonalize it with the formula $\Lambda = P^{-1}AP$, where $P$ is a block matrix whose columns are the eigenvectors of $A$? I ...
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1answer
21 views

Rank of a linear map $f: \mathbb{R}^n \to \mathbb{R}^m$ is equal to $m$?

Take a linear map $f: \mathbb{R}^n \to \mathbb{R}^m$, where $n>m$. Is the rank of $f$ always equal to $m$? Since the image of $f$ contains $\{f(a) | a \in \mathbb{R}^n \}$, the image will contain ...
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9 views

Gauss-Newton Non-Linear Squares Optimisation

I doubt this is solvable at all, but I thought I will give a try. Essentially I am trying to extend Gauss-Newton algorithm to 2nd Taylor term. ...
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1answer
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Linear dependence under transformation

I have a linear map $f:\mathbb{R}^6\rightarrow\mathbb{R}^4$ and I'm asked to show that if $u$,$v$ and $w$ are linearly dependent vectors in $\mathbb{R}^6$ then also $f(u),f(v),f(w)$ in $\mathbb{R}^4$ ...
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The annihilator induces a module

Let $R$ be a ring, and $M$ an $R$-leftmodule. Let $\operatorname{Ann}_R(M)$ be the annihilator of M, meaning that $r m = 0 \space\space\space\space \forall r \in \operatorname{Ann}_R(M), m \in M$. ...
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1answer
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Homomorphisms inbetween factor modules

Consider the Ring $\mathbb{Z}$ and the two ideals $(n), (m)$, where $n, m \in \mathbb{N}$, and consider $GCF(m, n)$ (the greatest common factor of $m, n$). Let p: $\mathbb{Z}/(m) \to \mathbb{Z}/GCF(m, ...
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Find vector and parametric vector of a line

I have a line that is perpendicular to a plane. This perpendicular line is $3i-2j+6k$. I've also been given that the line passes through $A(2,3,0)$. I'm unsure on how to represent this line as a ...
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How do I find the determinants of $3A, -A, A^2, A^{-1}$, where A is an $4\times 4$ matrix and $\det(A) = \frac{1}{3}$?

I am getting crazy with these determinants. For a little, I thought I could solve a problem alone, because I had understood more or less how to calculate the determinants of a matrix, but I am back to ...
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1answer
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Show that a positive definite (not necessarily symmetric) matrix induces a hyperellipse

Consider $A\in M_n(\mathbb{R})$ a positive definite matrix and a matrix $B\in M_{n \times p}(\mathbb{R})$, with $n\geq p$ and $rank(B)=p$. i) Show that $C=B^TAB$ is positive definite. ii) Show that ...
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Null space and Matrix equations

http://studyguide.pk/Past%20Papers/CIE/International%20A%20And%20AS%20Level/9231%20-%20Further%20Mathematics/9231_s03_qp_1.pdf I would like to know the method to answer question 8. I have been having ...
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1answer
52 views

What is the difference between $A^{-1}$ and $A^\Theta$?

Let $A$ be a square invertible matrix. Then $$A \cdot A^{-1} = I$$. Let $A^\Theta$ be the conjugate transpose matrix of $A$. Then $$A \cdot A^\Theta = I$$. Both on multiplication with $A$ gives ...
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Proving that a linear functional is matrix trace [duplicate]

Let $W=\operatorname{M}_{n\times n}(\mathbb{F})$ (square matrices $n\times n$ over $\mathbb{F}$), and $f\in W^*$. If $f(AB)=f(BA)$ for every $A,B\in W$ and $f(I)=n$ prove that ...
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How do I find a matrix for this linear transformation

The problem is that the transformation is defined by T(p)=(p(0), p(1) P(-1), p(0)) B is the standard basis for M22 and B' = {1, x, x^2}. How ...
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1answer
11 views

How to find linear equation from text

A company produces covers for cell phones. The costs of a production of $x$ units can be described by a linear function $C(x)=ax+b$. At a production of $150$ units the costs are DKK $6000$. At a ...
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Find the vector passing through a given point which is orthogonal to a given triangle in space

I'm given this problem where I have 3 points in space $A(3, -1, 2)$, $B(-2,1,2)$ and $C(2, 0, 5)$. I need to find the vector passing through point $A$ that is perpendicular to the triangle made by ...
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Change of basis and similarity

Consider the transformation $T$ in the standard basis: $$[T]_B\begin{bmatrix} 0&3&1 \\ -1&3&1 \\ 0&1&1 \end{bmatrix}$$ Also consider the two matrices: $$A_1 = ...
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$\dim (W_{1} \cap W_{2}) = \dim W_{1}$ implies $W_{1} \subset W_{2}$?

Let $V$ be a finite-dimensional vector space and let $W_{1}, W_{2}$ be subspaces of $V$. If $\dim (W_{1} \cap W_{2}) = \dim W_{1}$, must $W_{1} \subset W_{2}$? Since $\dim (W_{1} \cap W_{2}) = \dim ...
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Finding an Extermal of Hard Examples?

Who Can show me the calculation for solving extermal for $$\int_0^1 (x^2+ \dot {x}^2+2xe^t) dt \quad \text{ when }\quad x(0)=0,\;x(1)=free.$$ My TA say a short answer and I Couldn't reach to ...
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Find global minimum of the function

I need to find the global minimum of the function $$f ( x) = \langle Ax,x \rangle + 2\langle b ,x\rangle+c$$ where $c \in \mathbb{R}$ is constant, $b \in \mathbb {R}^n$, and $A$ is a positive ...
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generalized inverse and its applications

As generalized inverse has vast applications in the field of linear algebra, but why the generalized inverses is important? why we are studying about it?
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26 views

tensor of two vector space [on hold]

I don't know how to show this problem please help me. let $R$ be a domain and $Q=Frac(R)$ if either $C$ or $A$ is a vector space over $Q$,prove that both $C\otimes_RA$ and $Hom_R(C,A)$ are also vector ...
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1answer
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Proof of Hyperbolic Functions

Find the proof:  (a) Use the definitions cosh(x)= 1/2(ex +e^−x) , sinh(x)= 1/2(e^x − e^−x) to express sinh(x + y) and cosh(x + y) in terms of cosh(x), sinh(x), cosh(y) and sinh(y). (b) Using the ...
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1answer
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Problem with Broyden update: Divide by a matrix?

I am implementing a maximum likelihood method (the EM algorithm) for which I'm using Broyden's method at each iteration. Here is the formula: $\Delta A = \frac{(\Delta \theta - A ...
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50 views

An example of non euclidean inner product [on hold]

Please give me an example of non euclidean inner product.Is there any method to construct such an inner product?
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The dual of the dual, isomorphism, equivalence of functions

Problem: Let $V$ be a f.d. vector space. Define $\theta:V\to (V^*)^*$ given by $\theta(v)(\alpha)=\alpha(v).$ Let $T:V\to V$ and $T^*\ ^*:(V^*)^*\to(V^*)^*$ be linear maps. Prove $T=T^*\ ^*$. ...