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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2answers
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tangent plane for y^x at point (2,1)

I test my answer using wolfram alpha pro but it gets a different result to what I am getting. This is homework. My result is z= 2(y-1) partial derivative with respect to y is ...
2
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1answer
12 views

How to find the minimum sum of unknown variables that is a solution to a system of two linear equations?

I'm trying find the minimum sum of $x_{1} + x_{2} + ... + x_{n}$ where these are a solution to a linear system of two equations. System of linear equations in general form: $$ a_{11}x_{1} + ...
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0answers
17 views

algebra questions

On the weekend you played Rugby League. You scored three tries, made two conversions and one field goal. How many points did you gain for your team in total? Write a general equation for this problem. ...
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0answers
32 views

Number of matrices $A \in M_n(\mathbb{F}_q)$ where $A^2 = 0$.

What is the number of matrices $A \in M_n(\mathbb{F}_q)$ for which $A^2 = 0$ (as a function of $n$ and $q$)?
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5answers
28 views

Seemingly impossible problem involving linear combination of vector components.

Express $\langle 4, -8 \rangle$ as a linear combination of $\vec{u}$ and $\vec{v}$, given $\vec{u}=\langle 1,1 \rangle$ and $\vec{v}=\langle -1,1 \rangle$. So, I set up: $\vec{i}=\langle 1,0 \rangle$ ...
2
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2answers
34 views

Is $V$ a simple $\text{End}_kV$-module?

Let $V$ be a finite-dimensional vector space over $k$ and $A = \text{End}_k V$. Is $V$ a simple $A$-module?
2
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0answers
29 views

How to solve these equations?

How to solve these equations for a, b, c and x? I have the following: $ 2a+b+c = 1$ $a = (a+b)x + 0.25(a+c) $ $a=(a+c)(1-x)$ $b=a(1-x)+c(x-0.25)$ $c=b(1-x)+a(x-0.25)$ I tried, but ended ...
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0answers
14 views

Quadratic form in Hilbert space associated with orthogonal projection operator

we are in Hilbert space $L^2 $ and we are given subspace of dimension $2K$ $$ V=Vect\{ g_k,\bar{g_k},1\le k\le K \}$$ $V$ is a sum of $K$ subspaces of dimension 2 $$ W_k=Vect \{g_k,\bar{g_k} \} $$ now ...
4
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0answers
32 views

Square root of differentiation

Let $T=d/dx$ be the differentiation operator on vector space $V=C^{\infty}(\mathbb{R})$, the space of real (complex) valued smooth maps on real line. To what extent, all subvector space ...
1
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2answers
14 views

What am I doing wrong in finding the orthogonal projection of a vector onto the subspace V?

Let $V∈ℝ^5$ be the subspace $V=span{(2,0,0,0,1),(0,2,0,3,0)}$ and let $w=(0,0,-4,-1,-1)$. Find the orthogonal projection of $w$ onto V,using exact values in your answer. My Approach Let the ...
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1answer
15 views

Finding Marginal Density functions with $Y\sim N_4(\mu,\Sigma)$

Suppose $Y$ is $N_4(\mu, \Sigma)$ where $$\mu = ( 1,2,3,-2)'$$ and $$\Sigma =\begin{bmatrix} 4& 2& -1& 2 \\ 2& 6& 3& -2 \\ -1& 3& 5& -4 \\ 2& ...
3
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1answer
24 views

Prove $v,w\in Z_{p}\times Z_{p}$ is linearly independent when $p=2$ and dependent when $p=3$

I need to prove that $\{v=(6,9),w=(7,8)\}\in Z_{p}\times Z_{p}$ is linearly independent when $p=2$ and linearly dependent when $p=3$. The problem is my freshman algebra course did not cover rings and ...
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0answers
7 views

About finding the common diagonalizing similarity transformation.

Say I have $2k$ matrices $M_{a_1b_1}$, $M_{a_2b_2}$,..,$M_{a_kb_k}$ and their negatives. Here $M_{a_ib_i}$ is such that it has $0$ everywhere except that it has $1$ at $(a_i,b_i)$ and $(b_i,a_i)$ ...
2
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2answers
32 views

Prove function space is linearly independent.

Let $V$ the space of all funcions $f:Ŗ\rightarrow R$. Prove that the ten functions defined by $x\rightarrow |x-1|$,$x\rightarrow |x-2|$,....,$x\rightarrow |x-10|$ are linearly independent. I need ...
1
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1answer
10 views

Given $k$ distinct linear operators, prove such an $\alpha$ exists

I have $k$ distinct linear operators $\{\phi_i\}$ which act on $V$, a vector space on some number field $K$ (in the sense that $\Bbb Q$ is the smallest possible one). Now I have to prove that there ...
1
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1answer
7 views

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 ...
1
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2answers
55 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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0answers
23 views

Unitary matrix question [on hold]

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
24 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, ...
0
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0answers
12 views

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 ...
1
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1answer
31 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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0answers
17 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 ...
0
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1answer
23 views

Dice game and points [on hold]

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 ...
1
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1answer
34 views

An equivalent definition of the condition number of a matrix [on hold]

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 ...
0
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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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2answers
29 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 ...
3
votes
1answer
44 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
39 views

I need to calculate a Linear Algebra question on augmented matrix [on hold]

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 ...
0
votes
1answer
18 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 ...
0
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2answers
17 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 $S = T$?
0
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1answer
15 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 ...
0
votes
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, ...
0
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2answers
16 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) ...
0
votes
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} ...
0
votes
1answer
22 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 ...
4
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0answers
21 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 ...
6
votes
1answer
18 views

$AXB$ 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 ...
2
votes
1answer
17 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 ...
1
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2answers
57 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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0answers
11 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 $n-1$, ...
2
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0answers
14 views

dimension of intersection of subspaces

In a 13-dimensional vector space, the dimension of intersection of two 6-dimensional subspaces is at least 1 at most 1 at least 6 at most 6 My thought:For two subspaces U and W of vector space V, ...
4
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0answers
12 views

$GL_n(\mathbb{C})$-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 ...
3
votes
3answers
50 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
votes
3answers
51 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 ...
0
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1answer
14 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 ...
0
votes
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 ...
5
votes
2answers
35 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 ...
0
votes
1answer
20 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 ...
0
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0answers
20 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 ...