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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Recovering rectangular paper dimensions from picture of it.

I derived some transformations that allow me to map the subset of an image corresponding to a piece of paper in a picture to the entire image (as in ...
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28 views

Finding kernel and image of a Linear Transformation

Let $S:P_2(\mathbb{R}) \longrightarrow \{W \in M_2(\mathbb{R}) ~|~ A~ \text{is symmetric} \}$ given by the following rules $1+x \mapsto \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$ $1-x ...
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50 views

Given a CRS stored matrix A, provide an algorithm for calculating vector u.

Given an $NxN$ matrix $A$ and vectors $u,v,b$ such that: $$u_i = {\frac1{a_{ii}}}(b_i - \sum_{j=1,j\neq{i}}^n a_{ij}v_i)$$ And considering $A$ is stored using CRS, provide an algorithm (or ...
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31 views

Matrix Multipication and Inner Product

I am wondering about the difference between matrix multiplication and inner product. It is regarding the following question: Let $N \in M^{\Bbb C}_{n \times n}$ be a normal matrix. Prove that if ...
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4answers
555 views

Eigenvalues of reflection

Why are the eigenvales of a reflection $Rx=\rho x$ in a $n$-dimensional vector space just $\lambda=-1,1$? I can't seem to convince myself of this.
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2answers
31 views

How does one prove that two norms are equal if and only if their closed 1-balls are equal?

Let $X$ be a vector space and let $||\bullet||_1$ and $||\bullet||_2$ be two norms on $X$. I wish to prove that $||x||_1=||x||_2$ for all $x\in X$ if and only if $\{ x\in X \text{ such that ...
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22 views

Detecting eigenvals of opposite sign

Consider a large (in the region of 500 by 500 up to 2000 by 2000) real, square symmetric matrix. What would be a good way of algorithmically determining if it has any pair of eigenvals with opposite ...
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1answer
40 views

prove that transformation are non linear transformation. [closed]

Prove that the following transformation is not linear $T(x_1, x_2, x_3 )=( | x_1 | , x_2-x_3)$
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35 views

If $A$ is a rank one linear transformation, show there is a unique scalar $\alpha$ such that $A^2 = \alpha A$

If $A$ is a rank one linear transformation, show there is a unique scalar $\alpha$ such that $A^2 = \alpha A$. Then, if $\alpha \neq 1$, show that $1-A$ is invertible. This is problem 9, section ...
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1answer
32 views

In the LQ control problem why is $2PA = A^{T}P + PA$

I do not understand why the equation below holds assuming $A$ and $P$ are both square matrices in $\Bbb{R}^{n*n}$ and $P$ is symmetric and positive definite (i.e. $ P = P^{T} $ and $ x^{T}Px > 0 $) ...
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18 views

The subspace generated by $A\cup B$ is contained in the union of subspaces generated by A and B?

can you help me with this? Let V a vetorial space and A and B finite sets of V. If R is the subspace generated by $A\cup B$ and U and W are the subspaces gerenated by A and B, respectively, then ...
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3answers
65 views

Understanding of a proof about upper/lower triangular matrices property under multiplication

Show that the product of two upper (lower) triangular matrices is again upper (lower) triangular. I have problems in formulating proofs - although I am not 100% sure if this text requires one, as it ...
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1answer
31 views

Symmetric of equations (factorization)

Here is the question is there any tricks that I could use for this question. As I simply expand the whole equation and it is simply time consuming.
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1answer
98 views

Affine transformations - the meaning of contractivity

An affine transformation $\omega \colon \mathbb{R}^2 \to \mathbb{R}^2$ is a linear mapping followed by a translation, in other words $$ \omega(x) = Ax+t = \begin{pmatrix} a & b \\ c & d ...
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13 views

theorem of the alternatives and infeasibility

Trying to prove a simple thing using Motzkin (or Farkas) theorem. Given a set in two dimensions defined as $x: Ax \leq b$, we assume that $\begin{bmatrix} 1 & - 1 \\ A \end{bmatrix} \leq ...
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28 views

Convex minimization of a linear function

I have the following optimization problem to be solved $R^{k+1}$ = $\text{argmin}_R$ $\frac{1}{2\mu}\left\lvert R - W_{R}^{k}\right\rvert_{F}^{2}$ + $\frac{1}{2}\left\lvert R - ...
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72 views

Basis for the dual space of polynomials

I am stuck on the following question: Let P(n) be the space of real polynomials of degree at most n. For r ∈ ℝ define πᵣ ∈ P(n) * by πᵣ(p) = p(r). Show that π₀, π₁, ... form a basis for P(n) *, and ...
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111 views

A familiy of hermitian matrices with arbitrarily large expectations: is one of them positive definite?

Imagine that for a family of Hermitian $N\times N$ matrices $C(\alpha)$, $\alpha>0$, one knows that, for any $x\in\mathbb{C}^N\setminus\{0\}$, $\lim_{\alpha\to+\infty}\langle ...
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0answers
9 views

inverse and unitary invariance

I want to simplify the expression $(\mathbf{V}^H \mathbf{X} \mathbf{V} )^{-1}$, where $V \in \mathbb{C}^{M\times d}, M>d$ is unitary but not square, i.e., $\mathbf{V}^H\mathbf{V}=\mathbf{I}_d$ but ...
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1answer
31 views

some problems related to characteristic polynomial and rank.

Let $A$ be a real $4 \times 4$ matrix with characteristic polynomial $p(t)=(t^2+1)^2$, then $A$ is diagonalizable over complex numbers, but not over real numbers. Is this statement true or false? I ...
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15 views

A sesquilinear form is left non-degenerate if and only if it is right non-degenerate

Given a finite dimensional vector space $V$, how to prove that a sesquilinear form is left non-degenerate if and only if it is right non-degenerate, where a form $f$ is said to be left non-degenerate: ...
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2answers
42 views

Does injectivity imply that the components span the dual space?

Let $V$ be a finite-dimensional vector space. Let $f=(f_1,f_2,\dots)\colon V\to \mathbb{R}^{\mathbb{N}}$ be an injective linear map. Do $f_1,f_2,\dots$ span $V^*$? I know that the answer is yes when ...
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1answer
23 views

Proving orthogonal properties in $R^n$

Proposition 4. Suppose that $X\prec\Bbb R^n$ is a subspace of $\Bbb R^n$. Then (1) $X^\bot\prec\Bbb R^n,$ (2) $X\cap X^\bot=\{0\},$ (3) $(X^\bot)^\bot=X$. I'm trying to prove ...
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41 views

Jacobians and least squares normal equations

I am trying to solve a problem with non-linear least squares (Gauss-Newton), but the question is more about a single iteration, or least squares. The Jacobian $J$ for each constraint is a 1x6 vector. ...
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1answer
91 views

Difference between epimorphism, isomorphism, endomorphism and automorphism Linear transformations with examples

Can somebody please explain me the difference between Linear transformations which epimorphism, isomorphism, endomorphism or automorphism. I would appreciate if somebbody can explain the idea with ...
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1answer
60 views

Rank of a fat random matrix

Let $\mathbf{R} \in \mathbb{C}^{~n \times k} $ with $n \leq k $ be a random matrix, whose entries are i.i.d zero mean random variables with circularly symmetric Normal distribution. Two questions: ...
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33 views

Finding the determinant of a block matrix (Linear Algebra)

Let $A$, $B$ be $n$ by $n$ matrices. Let $0$ be the all zero matrix of size $n$. Show that $$\det\left[\begin{array}[cc]\\A& 0\\ 0&B\end{array}\right]=\det(A)\det(B)$$ What i tried What i ...
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2answers
79 views

Suppose $\lambda$ is an eigenvalue of $T$. Prove $\lambda = 2$ or $3$ or $4$

Suppose $T\in(V)$ and $(T-2I)(T-3I)(T-4I) = 0$. Suppose $\lambda$ is an eigenvalue of $T$. Prove $\lambda = 2$ or $\lambda = 3$ or $\lambda = 4$ What properties of polynomials will prove this?
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43 views

Approximate an exponential factor

What math methods can I use to approximate lambda in the following system of equations?: $$ e^{-0.05\lambda}=0.5469\\ e^{-0.1 \lambda} = 0.3229\\ ...\\ e^{-0.2 \lambda} = 0.1226$$ I am trying to fit ...
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30 views

Choose a basis for the space of polynomials of low degree, and extend it to a basis of a larger space

Choose a basis of a vector subspace and extend it to a basis of a vector space State dimensions of a vector subspace and a vector space $$W=\{f(x); f(x) \text{ is a polynomial, degree of $f$ is }≤ ...
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1answer
23 views

Find the least square ssolution of the linear system…

Hi I'm looking for the least squares solutions of.... $$ \begin{pmatrix} 1&-1 \\ -1&2 \\ -1&0 \end{pmatrix} \begin{pmatrix} x\\y \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \\ 5\end{pmatrix} ...
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2answers
24 views

Linear Algebra vector matrix problem

Link to the picture of the problem So, do I simply put the numbers in a augmented matrix and into row echelon form, and solve for m? The variable m throws me off.
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22 views

Upper and Lower Uni-triangular groups

Let $F$ be a field, and consider the $n\times n$ matrix groups $U$ and $L$ over $F$ as follows: $$ U=\begin{Bmatrix} \begin{bmatrix} 1 & * & \cdots & *\\ & 1 & \cdots & *\\ ...
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20 views

Polynomial relation with matrix coefficients

Lemma: Let $P$ and $Q$ be two polynomials with matrix coefficients, say $$ P(x) = \sum P_{m}x^{m} \hspace{5mm} Q(x) = \sum Q_{j}x^{j}.$$ Then the product $R(x) = P(x)Q(x)$ is then $R(x) = \sum ...
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1answer
19 views

What is the purpose of $v$ in the parametric equation for a sphere?

The longitude / latitude parameterization of a sphere is described by: $x = cos(φ) * cos(θ) \quad y = cos(φ) * sin(θ) \quad z = sin(φ)\quad$ where $\quadθ = 2 π u$ and $φ = π v - π / 2$ I ...
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1answer
23 views

Find all eigenvalues and eigenvectors of $P \in L(V)$ by $P(u+w) = u$

Suppose $V = U \oplus W$, where U and W are nonzero subspace of V. Define $P\in L(V)$ by $P(u+w)= u$ for $u \in U$ and $w \in W$. Find all eigenvalues and eigenvectors of P. Assuming all the notation ...
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1answer
39 views

Linear Algebra, Linear Operators

State if the given map is a linear operator from $V$ to $W$. Explain your answer. $V = W = \{f(x)\mid f(x) \text{ is a polynomial, degree of $f$ is } \le 5 \text{ and } f(1) = f′(1) = 0 \}$, ...
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1answer
55 views

Does $V=im(T)\oplus ker(T)$ hold if $T$ is an idempotent linear transformation? [closed]

This is a well known fact for $T$ is a linear transformation. Would it be any different $T$ is idempotent, $T^2=T$?
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1answer
14 views

Different bases for a subspace generated by a set of vectors

Let's say we have a set of vectors $A$, which is the basis for another vector space $B$. We let $a_1$ be any vector such that $a_1 \in A$. Now let's say we have a set $C_1 = \{ a_1 \}$. If $\dim(A) ...
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2answers
35 views

Linear Algebra, Linear Operator

State if a given map is a linear operator from V to W. Explain your answer. $V=R^2$, $W =R^3$, $f(a,b)=(a−b+1,−a,b)$ This is what I have thus far... $T(u+v) = T(u) + T(v)$ $T(αv) = αT(v)$ ...
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1answer
60 views

Find all real number x such that [2,-1,3] and [x,-2,1] are orthogonal

Find all real number x such that [2,-1,3] and [x,-2,1] are orthogonal. I saw an example that just simply used dot product $[2, -1, 3] \cdot [x, -2, 1]$ = $2x + 2 + 3 = 2x + 5$ The two vectors will ...
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26 views

Is this proof of Steiniz' lemma correct?

Our professor of linear algebra gave the following proof of Steiniz' lemma: Lemma: Let $V$ be a vector space over a field $K$. Suppose $v_1,...,v_m\in V$ are linear independent and $w_1,...,w_n\in V$ ...
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2answers
39 views

Inversion of a Block Matrix

Let $S$ to be a symmetric and positive semi-definite matrix of size $n$. What is the inverse of the following block matrix $$ M_{2n\times 2n}= \begin{bmatrix} aI+S & -I\\ -I & aI+S ...
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0answers
21 views

Writing Matrix with respect to bases

Let $T: \{A \in M_2(\mathbb{R})\mid A \text{ is symmetric} \} \longrightarrow \mathbb{C}$ be a Linear Transformation given by the following rules. $T\left( \begin{bmatrix} 1 & 0 \\ 0 & 0 ...
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0answers
25 views

SVM : Why can we set 1 in the hyperplane equation?

I am reading the Wikipedia article about SVM and there is something I don't understand. When they say: These hyperplanes can be described by the equations $$ wx - b=1 $$ and $$wx - ...
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2answers
35 views

Find the points of intersection of the following pairs of lines

$\begin{bmatrix}x \\y \\z\end{bmatrix} = \begin{bmatrix}3 \\-1 \\2\end{bmatrix} + t\begin{bmatrix}1 \\1 \\-1\end{bmatrix}$ $\begin{bmatrix}x \\y \\z\end{bmatrix} = \begin{bmatrix}1 \\1 ...
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1answer
79 views

Solving a partial sum…

Hey can anyone help with this? This is the classic NPV equation: $$\texttt{NPV = -CapEx} + \sum_{i=0}^n \frac{\texttt{Revenue − Costs}}{(1+\texttt{Discount})^i}$$ For my purposes all the elements ...
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1answer
25 views

Find the parametric equations of the following Lines

Find the parametric equations of the following lines. a) The line parallel to $\begin{bmatrix}2 \\-1 \\0\end{bmatrix}$ and passing through P(1,-1,3). My solution $\vec{P}$ = $\begin{bmatrix}1 \\0 ...
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1answer
147 views

$2 \times 3$ Null Space Matrix

Find all $2 \times 3$ matrices $A$ whose null space is the plane $3x − 5y − z = 0$. I know the vector for null space is $(3,-5,-1)$ but I'm not sure how to use that to find all the diferent $2 ...
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2answers
26 views

In each case, find $\overrightarrow {PQ}$ and $\overrightarrow{||PQ||}$

In each case, find $\overrightarrow {PQ}$ and $\overrightarrow{||PQ||}$ P(1,0,1), Q(1,0,-3) My question is what formula would i use to find $\overrightarrow{||PQ||}$? For $\overrightarrow {PQ}$ ...