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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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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11 views

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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15 views

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): $P = A + v*$, $t \in R$, where $v = B-A$, is the only straight line which contains A and B.
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13 views

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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3answers
16 views

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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25 views

norm of matrix 1 [on hold]

SHOW ∥A∥1=∥AT∥∞? i dont solve.....
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3answers
83 views

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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1answer
15 views

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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2answers
22 views

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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7 views

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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17 views

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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13 views

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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1answer
18 views

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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1answer
21 views

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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1answer
25 views

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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19 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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3answers
56 views

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
20 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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0answers
7 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
11 views

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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1answer
16 views

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
15 views

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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1answer
18 views

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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3answers
54 views

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
18 views

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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1answer
49 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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0answers
24 views

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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8 views

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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3answers
21 views

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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0answers
19 views

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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3answers
30 views

$\dim (W_{1} \cap W_{2}) = \dim W_{1}$ 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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0answers
25 views

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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0answers
18 views

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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6 views

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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24 views

tensor of two vector space

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
21 views

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
16 views

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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2answers
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 $T=T^*\ ^*$. ...
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1answer
17 views

Basis, polynomial vectors

Given the vector space $P_3(R)$, find a basis for it containing the polynomials $x^2 + 1$ and $x^2 - 1$. To find a basis, I need to find whether there exists constants in front of these two vectors ...
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2answers
31 views

Intuitive explanation why if $P$ is a subspace of linear space $L$, then $L/P$ is not a subspace of $L$

Is there an intuitive explanation of why: if $P$ is a subsppace of linear space $L$, then $L/P$ is not a subspace of $L$. I know that it is true, but it is counter intuitive to me.
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29 views

Which space it belongs to Hilbert, Banach or something else?

The question is related to the following two questions. The link: Understanding Eigenvector defines the problem at hand. The question is the in which space (Hilbert or Banach or something else) we ...
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0answers
14 views

LU growth factor applied to LDL of a Positive Semidefinite matrix

For a Positive Semidefinite matrix $A$, which we can decompose through $LDL$ decomposition as follows: $A=LDL^\text{T}$; how can we prove that for a decomposition $A=LU=L(DL^\text{T})$, the growth ...
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0answers
15 views

Induced map between quotient vector spaces well defined and linear?

Suppose we have vector spaces $V,W$ and subspaces $$ V\supset V'\supset V''\\ W\supset W'\supset W'' $$ suppose also that we have a linear map $A:V\to W$. What does it take for this map to induce a ...
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2answers
25 views

Is $(A \oplus B)^{\perp} = A^{\perp}\cap B^{\perp}$?

Is this true? $$(A \oplus B)^{\perp} = A^{\perp}\cap B^{\perp}$$ I am trying to prove this, but could not find a way. Any suggestions would be much appreciated. Thanks.
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1answer
37 views

Eigenvectors of derivative

I'm trying to consider how linear algebra relates to calculus. It seems to me that the only eigenvectors of the derivative operator on $\Bbb R$ are the functions $ce^{kx}$ for constants $c$ and $k$. ...
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1answer
19 views

Why do the 1's in Gauss Jordan RREF need to be along main diagonal and not other diagonal?

I've practiced G-J elimination and understand most of the algorithm insofar as it represents the different manipulations one can apply to a system of equations. However, when we're talking about ...
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1answer
21 views

Linear Algebra-Vector Subspaces Question [on hold]

Let $U=\{f\in C^1([-1,1],\Bbb R);y'=y+1\}$. Is $U$ a subspace of $C^1([-1,1],\Bbb R)$?
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28 views

Rates and Linear Equations [on hold]

The following question I found in an old high school textbook I bought in a second hand bookshop. The question is exactly as it appears in the text with no additional information. The answer, with no ...