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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Kernel Principal Component Analysis (PCA)

I learn kernel PCA from wikipedia. In this article, the eigen equation is \begin{equation} N \lambda \vec{\alpha} = \boldsymbol{K} \vec{\alpha} \end{equation} where $\lambda$ is the eigen value, ...
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2answers
48 views

Help solve ${{z}^{3}}=\overline{z}$ ($z\in \mathbb{C}$)

Me and my friend try to solve $${{z}^{3}}=\overline{z}$$ where $z \in \mathbb{C}$. My way to solve it was: $\operatorname{cin}(\theta )=\cos(\theta)+\sin(\theta)i$ \begin{align} & z=r ...
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Complimentary subspaces ($K$ and $L$) problem, where $K=ker(p)$ and $L=ker(q)$ with $p,q: U \rightarrow U$ linear maps.

I am struggling with solving the following question: Let $U$ be a vector space over field $F$ and $p,q: U \rightarrow U$ linear maps. Assume $p+q=id_U$ and $pq=0$. Let $K=ker(p)$ and $L=ker(q)$. ...
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Find the number of distinct real values of $c$ such that $A^2x=cAx$

Let $$A= \begin{pmatrix} 5 & -3 & 0 \\ -3 & 5 & 0 \\ 0 & 0 & 2 \end{pmatrix}$$ and $c$ be a real no. such that $A^2x=cAx$ for some non-zero vector $x$. Then the number of ...
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Finding a base from matrice subspace

$U,W$ are sub-spaces of $M^\mathbb{R}_{2x2}$ $$U=Sp\left\{\begin{pmatrix} 1 & 2\\ 4 & 1 \\ \end{pmatrix}, \begin{pmatrix} 1 & -1\\ 3 & 2 \\ \end{pmatrix}, \begin{pmatrix} 1 & ...
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1answer
25 views

Solving second order nonhomogeneous linear equation

So i have the equation $$\frac{d^2y}{dt^2} + y = \sin(t)$$ I know the first step is to find the corresponding homogeneous equation, which i think would be: $$r^2+1=0$$ giving real roots and therefore ...
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25 views

prove or disprove Composition of linear transformations is one-one

Let $T$ and $F$ be 2 linear transformations from $\Bbb R^n \to \Bbb R^n $.Then prove or disprove $T \circ F=0 \to T$ is one-one. $|TF|$$=0$ $\implies$ $|T|$$|F|$$=0$ $\implies$ either $|T|$=$0$ ...
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Find two matrices $A$ and $B$ such that matrix $AB$ that is invertible but $BA$ is not.

I am trying to find two matrices $A$ and $B$ such that matrix $AB$ that is invertible but $BA$ is not. Have you got any ideas of easy examples? Thank you!
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Half space representation of a convex polytope

We know that the half space representation of a polytope is given by: $Ax<b$. Consider a convex polytope in $\mathbb{R}^3_+$ with vertices given by the following set of points: ...
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38 views

Determine whether the following set is a vector space

Being pretty new to Linear Algebra, I am trying find whether the set given is a Vector Space or not: \begin{equation*} V = \{A\in M_{3\times3} : AA^{t} = -I\}. \end{equation*} I've tried to solve it ...
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I have to show $e_i \in r(T)$ for all $i\neq r$, $1\leq i \leq {n}$.

$D$ be a division ring and $n>2$ a natural number. $e_i$ denotes the element in $D^n$ whose$ (i,j)$-entires are zero. Let $T\in M_{(n-1)\times n}(D)$ such that $T=\begin{pmatrix} T_1 \\ T_2\\ ...
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1answer
13 views

Calculating the determinant of an interationmatrix

Let $C_\omega = (I-\omega D^{-1}L)^{-1}((1-\omega)I+\omega D^{-1}R)$ then $\det(C_\omega) = (1-\omega)^n$ (Where $C_\omega\in \mathbb{R}^{n\times n}$, $R$ is upper triangular, $L$ is lower ...
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3answers
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To Find the Nullity of a Linear Transformation …

If $V(\Bbb R) $ be the vector space of $2\times2$ matrices and $$M=\begin{pmatrix} 1 & 2\\ 0 & 3 \\ \end{pmatrix}$$ If $T:V(\Bbb R)\to V(\Bbb R)$ be a linear transformation defined by ...
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18 views

The relation between the algebraic dimensions of a vector space and its dual

Let $V$ be an (infinite dimensional) vector space over the field $\mathbb F (=\mathbb R$ or$ \mathbb C$). If $\alpha$ is the dimension of $V$, for some cardinal number $\alpha$, I want to know, what ...
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1answer
19 views

What is the relation between the algebraic dimensions of a vector space and its dual?

Let $V$ be an (infinite dimensional) vector space over the field $\mathbb F (=\mathbb R$ or$ \mathbb C$). If $\alpha$ is the dimension of $V$, for some cardinal number $\alpha$, I want to know, what ...
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0answers
18 views

Is it true that $\tilde{P} = D^{\dagger} P D$ has non-negative entries?

Consider a $n \times n$ stochastic matrix $P$ (i.e. non-negative rows sum to one). We are interested in the matrix $\tilde{P} = D^{\dagger} P D$, where $D$ is a $n \times k$ matrix which is ...
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How to express $[u,v,w]$ as a function of $\phi,\theta$ and the norms of the vectors?

$u,v$ are linearly independent and $w$ is a non-zero vector. Let $Angle(u,v)=\phi$ and $Angle(u \times v,w)=\theta$. Express $[u,v,w]$ as a function of $\phi,\theta$ and the norms of the vectors. ...
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Which of the following is true for the following linear transformations?

If $T_1$ and $T_2$ are linear transformation on $V_2(\Bbb R)$ by $T_1(a,b)=(0,a)$ and $T_2(a,b)=(a,0)$ , then which of the following is true 1) $T_1T_2=0$ 2)$T_1^2=T_1$ 3)$T_2^2=T_1$ 4)$T_1T_2 $ ...
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30 views

Subspace of $\mathbb{R}^3$: Stuck on closed under addition

$$S=\left \{ \begin{bmatrix} x_{1}\\ x_{2} \\ x_{3} \end{bmatrix} ; x_{1}^{2}+x_{2}^{2}=x_{3}^{2} \right \}$$ Closed under addition: Let $\vec{y}=\begin{bmatrix} y_{1}\\y_{2} \\ y{3} ...
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How to find a set of integer vectors (of length L) such that all its subsets with size L are linearly independent?

Given a number $M\geq L$, how to find a set of $M$ vectors in $\mathbb{Z}_{\geq0}^{L}$, say $S=\{\mathbf{a}_1,\cdots,\mathbf{a}_L\}$, such that: 1-All subsets of $S$ with size $L$ are linearly ...
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Finding ordered basis of a linear transformation so that its matrix representation is diagonal [on hold]

Define $T: M_{n\times n} \to M_{n\times n}$ by $T(A) := A^t$. Note that $T$ is a linear transformation with eigenvalues $1$ and $-1$ with the set of eigenvectors $\{A \in M_{n\times n} \mid A = A^t\}$ ...
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194 views

Show that 1 and -1 are the only eigenvectors of this linear transformation

Define $T: M_{n\times n}\to M_{n\times n}$ by $T(A):= A^t$. Note that $T$ is a linear transformation. Show that $1$ and $-1$ are the only eigenvalues of $T$. Let $\lambda$ denote an eigenvalue ...
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1answer
24 views

Using Gauss elimination to check for linear dependence

I have been trying to establish if certain vectors are linearly dependent and have become confused (in many ways). when inputting the vectors into my augmented matrix should they be done as columns or ...
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1answer
25 views

eigenvalues of a matrix $A$ plus $cI$ for some constant $c$

If $A$ is a $n \times n$ real matrix with eigenvalues $\lambda_1,\lambda_2,...\lambda_n$, how does one get the eigenvalues of the matrix $A$ + c$I$, where $I$ is the identity matrix and $c$ is a ...
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linear map $f:V \rightarrow V$, which is injective but not surjective

I am trying to find a linear map $f:V \rightarrow V$, which is injective but not surjective. I always thought that if the dimension of the domain and codomain are equal and the map is injective it ...
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Is there any relation Trace and Boundary?

I understand the trace is sum of diagonal elements of a matrix. Further the boundary I always perceive as a 'end points' of bounded domain. However on the link below: ...
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3answers
57 views

Find an arbitrary power of a lower triangular matrix of size $3\times 3$

Let $F$ be a field and let $A=\begin{bmatrix}a&0&0\\1&a&0\\0&1&a\end{bmatrix}\in\mathscr{M}_{3\times 3}(F)$. Show that ...
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1answer
15 views

Doubt on Kantorovich inequality. Equivalence of inequalities.

To prove de Kantorovich inequality (for that we suppose the matrix A symmetric and definite positive) I need to demonstrate the next exercise: Proof that $$(x^TAx)(x^TA^{-1}x) \leq \frac{(\lambda_1 ...
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2answers
23 views

Orthogonality and projections

1)Consider the vector space $\mathbb{R}^n$ with usual inner product. And let S the subspace generated by $u\in \mathbb{R}^n,u\neq 0$. Find the orthogonal projection matrix $P$ onto the subspace ...
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1answer
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How to prove this identity involving characteristic polynomials on both sides?

Suppose $A\in \Bbb C^{m\times n},B\in \Bbb C^{n\times m},m\ge n$, prove: $$\det(\lambda I_m-AB)=\lambda^{m-n}\det(\lambda I_n-BA)$$ I don't want to get into nasty determinant calculation. Instead, I ...
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1answer
17 views

Finding a matrix by using hermitian

$A=\left[ \begin{array}{ccc} 4 & 0 & 0 \\ 0 & 1 & i \\ 0 & -i ...
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Why does $\|A\|^2_2 \geqslant \|Av^k\|^2_2$ where $\lambda_k$ is the largest eigenvalue of $A^TA$

Could someone explain why: Let $\lambda_k$ be the largest eigenvalue of $A^TA$, then $$\|A\|^2_2 \geqslant \|Av^k\|^2_2$$ ($v^k$ is the eigenvector corresponding to $\lambda_k$) From a ...
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Let $A$ be a complex $2$ by $2$ matrix having distinct eigenvalues $a, b$. Show that $A^n =\frac{ a^n}{a - b}(A - bI) + \frac{b^n}{b - a}(A - aI)$.

Let $A\in\mathscr{M}_{2\times 2}(\mathbb{C})$ be a matrix having distinct eigenvalues $a\neq b$. Show that, for all $n > 0$, \begin{equation*} A^n =\frac{ a^n}{a - b}(A - bI) + \frac{b^n}{b - a}(A ...
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For what values of $h$ the following system is consistent? [on hold]

For what values of $h$ the following system is consistent? $$ \left\{ \begin{array} 3x_1+4x_2-8x_3=h, \\ -6x_1-5x_2=2, \\ x_1+x_2-x_3=1. \end{array} \right. $$
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5answers
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Proving any vector in $\Bbb R^n$ can be written on the form $x = u + v$

I'm having a hard time understanding the solution of this exercise. The exercise says: Let A be an $n\times n$ matrix so that $$A^2 = A$$ Show that every vector $x$ in $\Bbb R^n$ can be written as ...
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2answers
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Let $C(X)$ be the column space of $X \in M_{n \times p}$. Starting off the proof of $C(X) \cup C(X)^{\perp} = \mathbb{R}^n$

Let $C(X)$ be the column space of $X \in M_{n \times p}$. Prove or disprove the following statement: Every vector in $\mathbb{R}^n$ is in either $C(X)$ or $C(X)^{\perp}$ or both. I interpret ...
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What is mathematical structure?

When we have an isomorphism, between 2 groups or vector spaces let us say, then it is said to be structure preserving. An isomorphism exists when there is at least one mutually invertible morphism ...
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Is set of three linear equations with three unknown solvable?

I have the following set of linear equations with the unknowns $h, n, i$ which I would like to express as a function of my known quantities, $e, f, g$: $$ e = h - n\\ f = h - i\\ g = i -n $$ with ...
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How can you find a matrix given you know its kernel/nullspace?

Suppose we are given that $\phi : \mathbb{R}^4 \rightarrow \mathbb{R}^3$, and also that $\ker\phi$ is the span of $\{\begin{pmatrix} 1 \\ 0 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 2 \\ 1 \\ 0 \\ 1 ...
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Linear Operator with finite dimension

I'm involved with this exercise. I would greatly appreciate your help Let $V$ be a vector space of dimension $n$ over a field $F$. Let $T: V \rightarrow V$ a linear transformation whose image and ...
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1answer
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Determining whether a set is linearly independent.

I am currently trying to determine whether the following set is linearly independent: $u=(4,3,-2), v=(2,-6,7), w=(14,-12,17)$ It can be easily observed that $w=2u+3v$ and since w can be expressed in ...
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3answers
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Same eigen values giving 2 different eigen vectors

For the matrix below I am getting two eigen vectors for a single eigen value $$ \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \\ ...
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2answers
138 views

How to determine whether a set is a vector space or not?

I'm currently learning Vector Spaces and although I understand the definition of what a vector space is, I can't seem to be able to find the correct answers when doing some questions. I would even say ...
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1answer
22 views

all abelian groups with 625 elements with 24 elements of order 5

Let $R$ be a principal ideal domain, $p \in R$ a prime element and $M$ a finitely generated $p$-torsion module of the form $M = R/(p^{e_1}) \oplus \cdots \oplus R/(p^{e_t})$. Let $_pM = \{m \in M: p ...
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Singular values and trace?

Given that $X$ and $Y$ are positive definite matrices, how can I bound the singular values $\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?
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2answers
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Injective linear mapping maps every plane to a plane through the origin?

Why an injective linear mapping from $R^3 \to R^3$ maps every plane to a plane through the origin? I can not understand this. It says also that if the mapping is not injective, it maps some line to a ...
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38 views

Matrix problem, subspace

Suppose you are given a matrix A and have calculated an echelon form R of A. (Note: R is not assumed to be in reduced row echelon form.) Which of the following statements must be true? (Select all ...
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1answer
33 views

Is inequality $tr(A^{-1^T} B) tr(A^T B^{-1}) \leq constant$ correct?

I have the following optimization problem \begin{align} \min_{A} &tr(A^{-1^T} B)\cr \text{subject to} &x^T A x > 0 \cr & A_{ii}=1 \end{align} where $A$ and $B$ are some positive ...
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2answers
30 views

Is the following set of vectors in $\Bbb R^3$ linearly dependent?

I am using Anton's Elementary Linear Algebra book (8e) and trying to do exercise set 5.3, question 2a It gives the vectors $(4,-1,2)$, $(-4,10,2)$ and asks if they are linearly dependent . My final ...
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1answer
29 views

For a given matrix $X$, find two linearly independent vectors in $C(X)^{\perp}$.

Let $$X = \begin{pmatrix} 1 & 1 & 4 \\ 1 & 2 & 1 \\ 1 & 3 & 0 \\ 1 & 4 & 0 \\ 1 & 5 & 1 \\ 1 & 6 & 4 \end{pmatrix}$$ Is there an easy way to ...