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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Isolate Costs in NPV equation

Hey can anyone help with this? This is the classic NPV equation: NPV = -CapEx + ∑ (Revenue − Costs) / (1+Discount)^i The partial sum is from i = 0 to n years. For my purposes all the elements ...
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1answer
36 views

Prove that three points define a unique parabola

How do we prove that there is always a unique parabola (with equation $y=ax^2+bx+c$) that passes through 3 distinct points $P_1 (p_1,q_1), P_2 (p_2,q_2), P_3(p_3,q_3)$ ? If I choose to use matrices ...
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Unitary matrix question

Let $A=\begin{pmatrix}1&4\\ 2&5\\ 3&6\end{pmatrix}$. Find $B$ such that $B^*B=I$ (identity matrix) and $Im A= Im(B)$ I know that the $Im(A)$ is the set of all possible linear ...
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0answers
22 views

Checking if this set is a vector sp.

Question: Define a set $V=\{(x,y):x,y\in\Bbb R\}$. For any two elements $u=(u_1,u_2),v=(v_1,v_2)$ in $V$ and $t\in\Bbb R$, addition and scalar multiplication as, ...
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I don't understand this definition of vector positivity in my linear algebra text

I don't understand why they say that the magnitude of v is greater than or equal to zero and then go on to say the magnitude of v is equal to zero if the vector is equal to zero. Shouldn't they use ...
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1answer
28 views

How are vector space dimension and basis related?

How are vector space dimension and basis related? (I am new to these concepts and know little to nothing about linear algebra/advanced calculus.) Thank you in advance.
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12 views

Linear algebra of generalized complex geometry

Let $V$ be an n dimensional real vector space and $V^*$ be an its dual. We consider a maximal isotropic subspace $L$ included in $V \oplus V^*$ with the inner product $\langle , \rangle$, where ...
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1answer
20 views

Dice game and points

A dice game is played and when a round is won the player earns 9 points and when a round is lost, a player loses 4 points. After 15 rounds a player has 18 points, how many rounds did that player ...
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1answer
29 views

An equivalent definition of the condition number of a matrix

How can I prove that the condition number can't be expressed by $$\kappa(A)= \sup_{\lvert\lvert x \rvert \rvert=\lvert \lvert y \rvert \rvert} \lvert\lvert Ax\rvert \rvert/\lvert\lvert Ay\rvert ...
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1answer
17 views

Is the spectral radius of a matrix a convex norm of it?

I am wondering if the spectral radius of a matrix is may be some kind of a norm ($l_{\infty}$-norm?) of it and if that is convex. Any pointers to related ideas would be helpful too.
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27 views

Linear independent sets

Let $S_1\in\mathbb{R}^{n}$ and $S_2\in\mathbb{R}^{n}$ be two subspaces of $\mathbb{R}^{n}$ Suppose $x_1\in S_1$, $x_1\notin S_1\cap S_2$. $x_2\in S_2$, $x_2\notin S_1\cap S_2$. Show that $x_1$ and ...
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1answer
36 views

Elegant way to prove that the space must be infinite dimensional?

Let $F(S,V)$ be the set of all functions from S to a vector space V, assume that $V\ne\{0\}$, and that S contains infinitely many elements, then we must have that $F(S,V)$ is ...
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2answers
37 views

I need to calculate a Linear Algebra question on augmented matrix

Build the augmented matrix corresponding to the linear system of equations $$ \begin{align} 3x − 2y + z &= 10\\ x + 4y + z &= 3\\ 11z &= 32\\ \end{align} $$ How many solutions does this ...
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1answer
17 views

Matrix and eigenvalues question hints?

This is the homework I have done part a, b, but I don t have any idea how to do the rest $y = 5$ and $z = 12 $ Those are the eigenvalues of matrix $A$ For part c, and d, I've tried to put some ...
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2answers
16 views

Linear transformation problem to prove equality of functions

If $V$ and $W$ are vector spaces over a field $\mathbb{F}$ and $S$, $T : V 􀀀\to W$ are linear transformations, such that $\ker(T) = \ker(S)$ and $\mbox{im}(T) = \mbox{im}(S)$. Is $S = T$?
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1answer
12 views

Given a set of complex subspaces, find a set of disjoint subspaces such that every original subspace is the span of the union of some subset

Suppose $S$ is a set of subspaces of $\mathbb{C}^{n}$ for some integer $n$. I would like to find a set $T$ of disjoint subspaces (not just pairwise disjoint - is there a clearer word for this?), such ...
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1answer
9 views

Sum of the vectors from one fixed vertex to each remaining vertex of a regular polygon

I'm attempting to calculate the sum of the vectors from one fixed vertex of a regular m-sided polygon to each of the other vertices. It's for a study guide preceding my Linear Algebra exam tomorrow, ...
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2answers
15 views

Prove that $ \sum_{k=1}^T t_k f(x_k) \leq B \Rightarrow \min_{ k \in \{1, \ldots, T \} } f(x_k) \leq \frac{ B }{ \sum_{k=1}^T t_k } $

Suppose $f(\cdot)$ is a positive real function, with positive real coefficients $t_k$s, and we know: $$ \sum_{k=1}^T t_k f(x_k) \leq B $$ Can we prove that? $$ \min_{ k \in \{1, \ldots, T \} } f(x_k) ...
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2answers
34 views

Matrix multiplication question for beginners

Can please someone explain me how to get this result? I mean where the 10 came from the 2nd board I don't get it :/ $$\begin{pmatrix}1&2&6\\ 3&0&3\\ 1&1&4\end{pmatrix} ...
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1answer
21 views

How does permutation works in “multimatrices”?

I want to adequately define a $m\times n$ "multimatrix" that satisfies these properties: 0.A $m\times n$ multimatrix has $m\times n$ entries just like a normal matrix. It is the positions they occupy ...
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19 views

Sending unitary group to reals, second differential at critical points.

Let$$X = U(n) = \{B \in \text{GL}_n(\mathbb{C}): BB^* = I\}$$be the group of $n \times n$ unitary matrices over $\mathbb{C}$, viewed as a real manifold. Fix a diagonal $n \times n$ matrix $A$ with ...
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1answer
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$AXB$ sort of decomposition?

Let $f: M_n(\mathbb{C}) \to M_n(\mathbb{C})$ be a $\mathbb{C}$-linear map (not necessarily an algebra homomorphism). Do there exist matrices $A_1, \dots, A_d \in M_n(\mathbb{C})$ and $B_1 \dots, B_d ...
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1answer
16 views

If $N$ is nilpotent of index $n\geq 2$ but $N^{n-1}\neq 0$ then there's no $A$ such that $A^2=N$

Let $N\in M_{n\times n}^{\mathbb{C}}$ a nilpotent matrix of index $n\geq 2$. Prove: if $N^{n-1}\neq 0$ then there does not exist a matrix $A\in M_{n\times n}^{\mathbb{C}}$ such that $A^2=N$. My ...
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2answers
55 views

Matrix with all 1's diagonalizable or not? [on hold]

This is a followup to my question here. Let $A$ be the $n \times n$ matrix over a field of characteristic 0, all of whose entries are 1. Is $A$ diagonalizable?
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9 views

Under what condition given $(x_1, y_1\cdot r_1),…,(x_n, y_n\cdot r_n)$ we can interpolate polynomial $T$ that has specific random root?

We know given $(x_1, y_1),...,(x_n, y_n)$ we can interpolate a polynomial $P$ of of degree at most $n-1$. Let us define polynomial $P=(x-\beta)\cdot g(x)$, where degree of $P$ is at most $n-1$, ...
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0answers
6 views

dimension of intersection of subspaces

In a 13-dimensional vector space, the dimension of intersection of two 6-dimensional subspaces is 1. at least 1 2. at most 1 3. at least 6 4. at most 6 My thought:For two subspaces U and W of vector ...
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$GL_n(\mathbb{C})$-conjugation invariant $\mathbb{C}$-valued polynomial has certain form.

What is the easiest way to see that any $GL_n(\mathbb{C})$-conjugation invariant $\mathbb{C}$-valued polynomial function on $M_n(\mathbb{C})$ has the form $f(A) = F(p_0(A), \dots, p_{n-1}(A))$ for ...
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3answers
49 views

Are vectors $e_i$ linearly independent if and only if the matrix $A$ is nonsingular? [duplicate]

Let $V$ be an inner product space over $\mathbb{R}$ or $\mathbb{C}$, and let $e_1, e_2, \dots, e_n$ be a collection of vectors in $V$, not necessarily orthonormal. (Here, $n$ has nothing to do with ...
2
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3answers
50 views

Eigenvalues of matrix with all $1$'s. [on hold]

Let $A$ be the $n \times n$ matrix over a field of characteristic 0, all of whose entries are 1. What are the eigenvalues of $A$, counted with their multiplicities?
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1answer
11 views

Proof for the fact that the method of characteristic equations is a valid method to solve recurrence relations

Could someone kindly point me to a proof of the fact that the method is characteristic equations is a valid way of solving recurrence relations? It seems fairly arbitrary to me. I would be grateful ...
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11 views

Project point on plane - Unique identfier?

I have a number of planes (in $\mathbb{R}^3$), each represented by a point $\vec{P_i}$ which lies within each plane and the normal vector $\vec{n_i}$. If I project a point $\vec{Q}$ (which does not ...
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1answer
23 views

Is a matrix similar to its RREF?

Let a matrix be denoted by A and its RREF be denoted by R. Then, is it true that R is similar to A? I am trying to find out Jordan canonical form of a large matrix. If I can somehow prove that a ...
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2answers
32 views

If $N$ is nilpotent then there exists $A$ such that $A^2=I+N$

Suppose $N\in M_{3\times 3}^{\mathbb{C}}$ is a nilpotent matrix. Prove that there exists $A\in M_{3\times 3}^{\mathbb{C}}$ such that $A^2=I+N$. Hint: find $A$ in the form $A=P(N)$ where $P$ is a ...
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1answer
18 views

Is Bs(1,-1) linear?

I would like to prove that the Baumslag-Solitar group $BS(1,-1)=\langle a,b| bab^{-1}=a^{-1}\rangle$ is embeddable in $GL_n(\mathbb{Z})$ for some nonnegative integer $n.$ So i tried to find two ...
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0answers
16 views

about the complex conjugation of a complex vector space $V\left(\mathfrak{\mathcal{C}}\right)$

I am a little confused about the complex conjugation of a complex vector space $V\left(\mathfrak{\mathcal{C}}\right)$. From other answers (Is a complex vector space closed under complex ...
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1answer
29 views

Minimal Polynomial

I have found the following characteristic polynomial: $$(x+2)(x-2)^2$$ I need to write down all the possible minimal polynomial, so I wrote: $${(x+2),(x-2),(x+2)(x-2),(x+2)(x-2)^2,(x-2)^2}$$ Why is ...
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26 views

Looking for references on the complexity of computation of a basis transformation matrix

I'm looking for some references on the complexity for the following kind of problem: Given two Basis $(a_1, ... ,a_n)$ and $(b_1, ..., b_n)$ of the $K(x)$-vector space $V$ I want to compute the ...
2
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1answer
48 views

$A^k = I$ implies diagonalizable? [duplicate]

If $A$ is a square complex matrix with $A^k = I$ (where $I$ is the identity matrix of the same size as $A$) for some positive integer $k$, does it follow that $A$ is diagonalizable?
5
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3answers
58 views

$\mathbb{C}$-algebra automorphism of $M_n(\mathbb{C})$ has form $X \mapsto AXA^{-1}$.

As the title suggests, what is the easiest way to see that any $\mathbb{C}$-algebra automorphism of $M_n(\mathbb{C})$ has the form $X \mapsto AXA^{-1}$ for some fixed $A \in GL_n(\mathbb{C})$?
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1answer
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Prove $2a^Tb \leq \|a \|^2 + \|b\|_*^2$ with dual norm

We know we can easily (**) prove $$ 2a^Tb \leq \|a \|_2^2 + \|b\|_2^2, \forall a,b $$ Is there a way to prove the following: $$ 2a^Tb \leq \|a \|^2 + \|b\|_*^2, \forall a,b $$ where ...
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12 views

norm on $\mathcal{B}(H)$

Given a Hilbert space $H$ and $a$ be a real numbers $\geq‎‎‎ 1$ , let $S_1(H)$ denote the space of trace-class operators on $H$, with the trace-class norm or Schatten 1-norm. That is $$ \Vert T ...
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2answers
38 views

Finding eigenvalues and eigenvectors of $2 \times 2$ matix

I having a few issues finding the eigenvectors for the following matrix: $$ \begin{bmatrix} -1 & -1\\ 0 & -2 \\ \end{bmatrix}$$ I calculated the eigenvalues to be ...
4
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1answer
23 views

Exist basis, simultaneously upper-triangular?

Let $A, B \in M_n(\mathbb{C})$ be such that $\text{rank}(AB - BA) \le 1$. Does there exist a basis of $\mathbb{C}^n$ with respect to which $A$ and $B$ are simultaneously upper-triangular?
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1answer
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For a linear system, why is direction “stored” in the variables when considering it as linear equations, but in vectors when its as a vector equation?

Given an arbitrary system of equations, why is direction in space "stored" in the variables when considering the system as linear equations, but "stored" in vectors when considering the system as a ...
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1answer
33 views

Is the free abelian group of rank 2 linear?

Is the group $\mathbb{Z}^2$ linear? By linear I mean There is a injective homomorphism from $\mathbb{Z}^2$ to $GL_n(\mathbb{Z})$ for some nonnegative interger $n.$ I tried the following homomorphism ...
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2answers
34 views

$f$ is a differentiable map and compute $Df(A)(H)$.

Let $f : GL(n, \Bbb R) \to GL(n, \Bbb R)$ be defined by $f(A) = A^{-1}$ where derivative of the matrix $A$ exists. Then $f$ is a differentiable map and compute $Df(A)(H)$. $A A^{-1} = I \implies ...
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1answer
27 views

If $A \in M_n(R)$, with $R$ a P.I.D., can $A$ be put in Jordan form iff all the roots of the characteristic polynomial are in $R$?

If $A \in M_n(R)$, with $R$ a P.I.D., can $A$ be put in Jordan form iff all the roots of the characteristic polynomial are in $R$? If this is false in general, is it possibly true for nilpotent ...
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1answer
31 views

Peculiar family of apparently positive semidefinite matrices

Let $x_1, \dots, x_n > 0$ be positive real numbers. From numerical experiments, it appears that the $n \times n$ matrix $$A_{ij} = \frac{1}{x_i + x_j} $$ is always positive semidefinite. Is ...
2
votes
3answers
40 views

Basis for subspace in $\mathbb{R}^4$

How would I start to answer this: Show that the vectors $(1,0,0,1)$, $(0,1,0,1)$, and $(0,0,1,1)$ form a basis for the subspace $V$ of $\mathbb{R}^4$ which is defined by the equation ...
0
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1answer
24 views

$m \times n$ matrix gives rise to a well-defined map from $\mathbb{R}^n$ to $\mathbb{R}^m$?

As the title suggests, how do I see that an $m \times n$ matrix gives rise to a well-defined map from $\mathbb{R}^n$ to $\mathbb{R}^m$?