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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Proving $AD_1A^{-1}=D_2$

I want to prove that if $A$ is a permutation matrix, and $D_1$ is diagonal, than $AD_1A^{-1}=D_2$ where $D_2$ is also a diagonal matrix. I have worked out that $A^{-1}=A^T$ and I can see that the ...
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1answer
9 views

linear transformation question (vector)

Not sure how to answer this question, please help!
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10 views

Different Inverse Approach

As it is known, we use inverse (Gauss Elm, Jordan...) or pseudo-inverse methods (LU, SVD, Chol, QR...) to solve linear equation namely $ A*x=b$ when $A$ is $[n,n]$ and $b$ is $[1,n]$ matrix. These all ...
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12 views

centralizers of $X= (M_2(F_p),+)$ in $H=GL_2(F_p)$

I need to find the centralizers of $X= (M_2(F_p),+)$ in $H=GL_2(F_p)$ in order to find the action of $H$ on $X$ which will help me find the orbits of $X$ I Know that the centralizers of $M_2(F_p)$ ...
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0answers
11 views

norm of symmetric positive definite matrix

How to prove this? I tried by using the fact that positive definite matrix is diagonalizable by orthogonal matrix and it preserve two norm. But I think this way dosen`t work.
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2answers
21 views

Find a and b such that the matrix is diagonalizable

Find a and b such that the matrix $$ \left( \begin{array}{ccc} 1 & a \\ 0 & b \\ \end{array} \right) $$ is diagonalizable. I know that $$ D = S^{-1} A S $$ where S is a matrix made of the ...
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2answers
16 views

Condition of distinct eigenvectors?

I am looking at this wikipedia page http://en.wikipedia.org/wiki/Matrix_decomposition#Eigendecomposition ...
0
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2answers
27 views

linear transform of functions

I'm trying to understand what a vector of functions is, from trying to understand how to solve linear homogeneous differential equations. It seems that functions can be manipulated as vectors as ...
0
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1answer
6 views

error in approximation a monotonic function in L^1

I am trying to solve this problem, but I am getting an incorrect solution. Here is the problem and my approach: Find the best $L^1$ linear approximation of $e^x$ on [0,1] i.e. minimize ...
0
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0answers
24 views

Linear transforms of functions [on hold]

if y is a vector; y = a.sin(x) + b.cos(x), a and b scalar then how is it that for any value of x, y is always a scalar value? Does this mean sin(x) is a vector, and sin(45) is not?
2
votes
2answers
26 views

Find value of k for distinct eigenvalues

Consider the matrix $$ A = \left( \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ k & 3 & 0 \end{array} \right) $$ where k is an arbitrary constant. For which values of k does A ...
2
votes
1answer
26 views

Use determinants to calculate the area bounded by 3 vectors

I have seen the proof of why the area of the parallelogram created by 2 vectors $u = \left(\begin{matrix} u_1\\ u_2 \end{matrix}\right)$ and $v = \left(\begin{matrix}v_1 \\ v_2 \end{matrix}\right)$ ...
0
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1answer
18 views

Solutions of a linear system writen as linear combinations of vectors [on hold]

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 ...
0
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0answers
30 views

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 ...
4
votes
2answers
51 views

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 ...
1
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2answers
35 views

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 ...
0
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1answer
15 views

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 ...
0
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0answers
14 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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0answers
5 views

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 ...
0
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2answers
27 views

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| ...
1
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1answer
23 views

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: ...
0
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0answers
16 views

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 ...
0
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1answer
22 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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0answers
34 views

Linear Algebra - Straight line determined by two distinct points [on hold]

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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0answers
18 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 ...
0
votes
2answers
22 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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0answers
31 views

norm of matrix 1 [on hold]

SHOW ∥A∥1=∥AT∥∞? i dont solve.....
2
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3answers
102 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 . ...
0
votes
1answer
16 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 ...
0
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3answers
33 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, ...
0
votes
0answers
14 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 ...
-1
votes
0answers
29 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 ...
3
votes
0answers
18 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 ...
2
votes
1answer
25 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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votes
1answer
22 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 ...
0
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2answers
32 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 $\begin{bmatrix}0 & X\end{bmatrix}$ ...
0
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0answers
21 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 $B^*=\{g_1,g_2,...,g_n\}$ ...
4
votes
3answers
60 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 ...
1
vote
1answer
23 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 ...
1
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0answers
10 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. ...
0
votes
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 $\mathbb{R}^4$ ...
0
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1answer
19 views

The annihilator induces a module [duplicate]

Let $R$ be a ring, and $M$ an $R$-leftmodule. Let $\operatorname{Ann}_R(M)$ be the annihilator of M, meaning that $r m = 0 \space\space\space\space \forall r \in \operatorname{Ann}_R(M), m \in M$. ...
1
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1answer
20 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, ...
0
votes
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
56 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 ...
1
vote
1answer
19 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
15 views

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
55 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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0answers
9 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 ...