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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Show that $\{1, \sqrt{2}, \sqrt{3}\}$ is linearly independent over $\mathbb{Q}$.

My apologies if this question has been asked before, but a quick search gave no results. This is not homework, but I would just like a hint please. The question asks Show that $\{1, \sqrt{2}, ...
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48 views

Prove that $R(P) = N (I − P) = X$ and $R(I − P) = N (P) = Y$.

Suppose that $V = X ⊕Y$, and let $P$ be the projector onto $X$ along $Y$. Prove that $R(P) = N (I − P) = X$ and $R(I − P) = N (P) = Y$. I know that from $V = X ⊕Y$ I got $v=x+y$ for $v,x,y$ are ...
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4answers
66 views

Angle between $\vec a$ and $\vec b $.

We got the same size vector $\vec a$ and $\vec b $. We know that the vector $\vec a +2\vec b$ and $5\vec a-4\vec b$ are perpendicular? $(\vec a +2\vec b) \perp (5 \vec a-4\vec b)$ What is angle ...
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1answer
115 views

An equivalent condition for a real matrix to be skew-symmetric

$A$ is an $n \times n$ real matrix. prove that $$A=-A^T \iff AA^T=-A^2$$. Thanks.
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1answer
125 views

Solution to this linear equation system

So this is my homework : Let $$ A= \begin{bmatrix} 1 & 0 & 1 & 3 \\ 2 & 0 & \lambda & 6 \\ 1 & 1 & 1 & 1 \\ \end{bmatrix} ...
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255 views

Find Rotation Matrix to rotate axes and move coordinates of point from P0 to P1

I have a point $P_0 = [x_0, y_0, z_0]'$. I want to rotate the axes so that the new coordinates will be $P_1 = [x_1, y_1, z_1]'$. Define the following rotation matrices: $R_x = \left[\matrix{ ...
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3answers
124 views

Map to symmetric matrices is surjective.

Let $M_{k,n}$ be the set of all $k\times n$ matrices, $S_k$ be the set of all symmetric $k\times k$ matrices, and $I_k$ the identity $k\times k$ matrix. Suppose $A\in M_{k,n}$ is such that ...
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1answer
97 views

Determinants and cofactors?

My professor gave us this definition for determinants for a $n \times n$ matrix $A$: $$\det(A) = a_{11}C_{11} + a_{12}C_{12} ... + a_{1n}C_{1n} $$ where $C_{1j}$ is the cofactor of $A$ on $a_{ij}$. ...
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36 views

Set of orthonormal vectors can be identified with set of matrices

Let $V_k(\mathbb{R}^n)$ be the set of all orthonormal $k$-tuples of vectors $v_1,\ldots,v_k\in\mathbb{R}^n$. Let $M_{k,n}$ be the set of all $k\times n$ matrices. Let $W=\{A\in M_{k,n}\mid ...
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1answer
34 views

linear dependncy of a random vector with respect to a reduced row echelon form in a finite field

Given a matrix with elements from a finite field $\mathbb{F}_q$, $A\in\mathbb{F}_q^{N\times M}$, where $q$ is the size of the field, $N<M$. Suppose that $A$ in the reduced row echelon form. ...
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238 views

$QR$ decomposition of rectangular block matrix

So I am running an iterative algorithm. I have matrix $W$ of dimensions $n\times p$ which is fixed for every iteration and matrix $\sqrt{3\rho} \boldsymbol{I}$ of dimension $p\times p$ where the ...
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1answer
74 views

Transform OR clause to algebraic equations (linear programming)

So basically my question is: does it exist a way to transform the clausure (a or b or c) into one or more algebraic equations giving as a result 0 or 1 AND that can be included in a linear programming ...
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4answers
106 views

Non-negative, real matrix $\Rightarrow$ non-negative, real eigenvalues?

Does a matrix with all non-negative, real entries have all non-negative, real eigenvalues? Where might I find a proof of such? Ideas: Perhaps we can multiply a prospective eigenvector so its biggest ...
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3answers
4k views

How to find a transformation matrix, given coordinates of two triangles in $R^2$

I am an undergraduate student, and today I was given two triangles, $T_1$ (green) and $T_2$ (blue) in $R^2$: I was then asked to find the transformation matrix transforming $T_1$ to $T_2$. What I ...
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99 views

A Gaussian as a mixture of two Gaussians

I'm quite convinced that a mixture of two Gaussians is itself a Gaussian iff mean and variance all equal : $N(x,\mu_0,\sigma_0) = aN(x,\mu_1, \sigma_1) + (1-a)N(x,\mu_2, \sigma_2) \iff \mu_0 = \mu_1 ...
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0answers
68 views

homework: rings, matrices and polynomials

$A,B$ are both $n \times n$ and diagonal matrices. Prove that there is a matrix $X$ which is $n \times n$, and polynomials $p$ and $q$ such that $A= p(X), B= q(X)$ Is this true for ANY 2 matrices (we ...
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138 views

Dimension of Image of Composition of Linear Transformations

Take two linear transformations T from V to W and S from W to U. I want to show that the dimension of the image of their composite SoT from V to U is 'smaller' than or equal to the dimension of the ...
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2answers
94 views

Calculating percentage to compensate for percent discount.

Missing something very basic here and cannot pin point it. We need to charge a client \$100 for a product. Let's say our payment processor charges us 10% on every transaction. We make this ...
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1answer
322 views

Condition Number of a block Matrix

Is this hypothesis true? $$cond([A,B])≤cond(A)+cond(B)$$ where $cond$ is the Condition Number. And is this true for rectangular matrices($nxm$)? Let's consider $3$ different conditions for $A$ and ...
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79 views

Linear Algebra - Kernel and Linear Transformation

Let $V $ be an n-dimensinal complex vector space, and let $\ T: V \rightarrow V $ be a linear transformation. Given that $$K_i = Ker \ T^i$$ Have shown that $ K_i \subseteq K_{i+1} for \ each\ i $ ...
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43 views

Inferenecs from the given: x = nonhomogenous + homogenous solution solve Ax = b. (GStrang P161, Ex 3.4B.4)

Given: All solutions to $\mathbf{Ax = b}$ have the form $\mathbf{x} = (1, 1, 0)^T + c(1, 0, 1)^T$. $\Large{\color{red}{1. [}}$ Then $A$ must have $n = 3$ columns. $\Large{\color{red}{]}}$ ...
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86 views

homework about homomorphisms : find all the homomorphisms

Find all the continuous homomorphisms $T:\mathbb{R} \rightarrow \mathbb{R}$ Find all the homomorphisms $T:\mathbb{C} \rightarrow \mathbb{C}$ (complex field) such that $T(x)=x$ for every $x$. ...
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1answer
73 views

Can the elements of a direct sum be thought of like that?

I've asked here about the tensor algebra, and I think that my problem is being able to realise the elements of a direct sum as linear combinations. Indeed the rigorous definition I have of the direct ...
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1answer
153 views

Richardson Iteration

Given the Richardson Iteration, $x_{n+1} = x_n + \alpha(b-Ax_n)$ (with $\alpha$ a scalar constant). To which polynomial $p(A)$ at step $n$ does this iteration correspond to? My first idea ...
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56 views

Generalized minimal residuals: eigenvalues and sets of functions

Can someone help me on this exercise (2 parts)? Thanks! Suppose that $S \subseteq \mathbb{C}$ is a set whose convex hull contains $0$ in it's interior (so $S$ is contained in no half-plane ...
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73 views

Find an SO(3) matrix which satisfies some linear constraints

I have the following optimization problem: $\displaystyle \min_R \sum_{i=1}^n (X_i^T R Y_i)^2$ where $R \in \text{SO}(3)$, i.e. is a 3x3 rotation matrix, and $X_i,Y_i \in \mathbb{R}^3$. If $n \le ...
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128 views

Are self-inverse operators normal?

Let $\mathcal{H}$ be an Hilbert space. Consider a bounded Operator $T:\mathcal{H}\to \mathcal{H}$. Suppose $TT=1$, does it hold, that $T^{*}T=TT^{*}$? If so, how does one show this? If not, what kind ...
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56 views

Where V is a Vector Space, $\forall \overrightarrow{v} \in V, 0\overrightarrow{v} = \overrightarrow{0}$

I'm denoting all vectors as such: $\overrightarrow{v}$. Any variable without an arrow above is a scalar. Suppose $V$ is a vector space over $F$, with additive identities $\overrightarrow{0}$ and $0$ ...
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59 views

Representing everywhere a camera can see as a matrix

I'm learning about Computer Graphics and there is one point really puzzling me. I understand that vertices (vectors) represent points in space and that transformation matrices represent changes that ...
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1answer
56 views

The determinant of a linear transformation on a finite vector space

Given a finite vector space $V$ and a linear transformation $f : V \rightarrow V,$ is it true that for any two ordered bases of $V$, call them $a$ and $b$, the determinant of the matrix of $f$ with ...
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2answers
60 views

How to find this Linear Transformation

Q. Find the Linear Transformation $T:V_3\rightarrow V_3$ , such that $T(0,1,2)=(3,1,2)$ $T(1,1,1)=(2,2,2)$ I tried considering $(0,1,2),(1,1,1)$ as basis, it doesnt seem to work that way. Just need ...
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1answer
189 views

If $M$ is positive definite, then $\operatorname{det}{(M)}\leq \prod_i m_{ii}$

In the Wikipedia article on positive definite matrices they claim that if $M$ is positive definite, then the determinant of $M$ is bounded by the product of its diagonal entries. How might we show ...
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106 views

A matrix-multiplication random walk

Let $x \in \mathbb{R}^n$. Consider an $n\times n$ matrix $A$. Suppose we're interested in how $||A^nx||$ grows with $n$, the answer (excluding pathological cases) is that it scales exponentially with ...
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60 views

$SO(3)$ with minimal and maximal trace.

Let $O(3)$ be the set of $3 \times 3$ orthogonal matrices. Let $SO(3)$ be a subset of $O(3)$ such that det($A$)=1 for all $A \in SO(3)$. Show that there is a matrix with minimal trace in $SO(3)$ and ...
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1answer
85 views

Cholesky decomposition: any theoretical value?

Just read the Wikipedia article on Cholesky decomposition. All the applications listed there were numerical. Are there theoretical arguments to which it is important? For instance, here there is an ...
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1answer
49 views

Help me to prove $\operatorname{Span}(X)=F$

Let $X \subset F$ be a subset with the following property: every linear transformation $A:E \rightarrow F$ whose image contains $X$ is surjective. Prove that $\operatorname{Span}(X)=F$. my doubt: ...
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2k views

Show that two matrices with the same eigenvalues are similar

First assume that $A$ and $B$ are $p \times p$ matrices and that $\lambda_1,... , \lambda_p$ are distinct eigenvalues of $A$ and $B$. I want to show that $A$ and $B$ are similar. Here is my ...
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52 views

eigenvalues of $AB$ are eigenvalues of $\sqrt{B} A \sqrt{B}$

Suppose $A,B$ are symmetric positive definite matrices. An author claims that the spectrum of $AB$ is the spectrum of $\sqrt{B}A\sqrt{B}$. Why? Certainly they have the same trace by cyclic ...
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1answer
70 views

On the extension of fields

Let $F\subseteq K$ be a finite field extension and let $a_1,..., a_n$ be an $F$-basis for $K$. I want to show that the matrix $A := (tr(a_ia_j))$ is singular if and only if $tr K =0$. Any suggestion ...
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29 views

Homogenous equation with only one solution

Is there a 2x3 homogenous equation with only one solution? I've tried looking for one but I am almost convinced that there is no such.
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73 views

How to prove this matrix identity?

This was a single step in a derivation, so I'm assuming there is a way to "see" this without writing down the expression for each entry: $$\sum\limits_{i=1}^n \left(x_i-\frac{1}{n}\sum\limits_{i=1}^n ...
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1answer
318 views

Geometric Interpretation of Jacobi identity for cross product

Is there a geometric "reason" for the Jacobi identity for cross products? Some geometric equality of some area ...? All proofs I know work by some form of linear algebra (or use the interpretation as ...
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1answer
79 views

Bra-ket multiplication

I'm studying a little bit of bra-ket notation and I found this property: $$\langle n| H_1 H_2|m\rangle=\sum_{k} \langle n|H_1|k\rangle \langle k|H_2|m\rangle$$ Is this property true? Why? Thank you! ...
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1answer
182 views

On eigenvalues, hermitian matrices and SVD

Are my ideas on the following "true or false"-statements correct? If $A$ is hermitian and $\lambda$ is an eigenvalue of $A$, then $|\lambda|$ is a singular value of $A$. My answer would be ...
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1answer
520 views

Does a point lie on a line with a parametric equation

Does the point $(0, 5, 5)$ line on the line with the parametric equations: $x = 3 - t\\y = 2 + t\\z = 2 + 2t$ This is the first time I see one of these, right now I assume it is as simple as solving ...
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105 views

Line segment intersection derivation

The book Graphics Gems III has the following algorithm: ...
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37 views

How do you know what a 3-variable equation represents geometrically or physically?

Example equation: $$z^2+xy-2x-y^2=1$$ I know that equation represents a plane. Is there any easy way to tell what a 3-variable equation represents from just looking at it?
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67 views

How to convert parametric form to a single algebraic equation?

I'm pretty sure this is impossible to do but here is my attempt. Parametric form: $$x=1+t\\y=2+2t\\z=3+3t$$ Attempt: $$(x,y,z)=(1+t,2+2t,3+3t)$$ That didn't really get me anywhere, so here I ...
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1answer
20 views

Dimension of a vector space-regarding

I want to find the dimension of the vector space $V=\{u\in \mathbb{R}^3:Mu^{t}=u^{t}\}$, where $M=\begin{pmatrix} 1&0&0\\ 0&\cos \theta& -\sin \theta\\ 0& \sin \theta& \cos ...
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75 views

Simple question about parametric equations of a plane in 3D

I'm quite rusty in Linear Algebra. If you have a plane in 3D with the equation $z=2$, what does $x$ and $y$ equal? Does $x=t$ and $y=t$? Because if I graph that in Wolfram Alpha, I don't get a ...