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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For $u,v$ nonzero, show $u\cdot v = 0$ implies $u$ and $v$ are linearly independent. [duplicate]

I know that if $u\cdot v = 0$ then by definition, $u_1v_1 + u_2v_2 + \cdots + u_nv_n = 0$ I also know that if $u$ and $v$ are linearly independent and a matrix $A$ has $u$ and $v$ as its successive ...
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95 views

almu and gemu of orthogonal projections and reflections?

Let Vbe an m-dimensional subspace of R^n. I have already found the eigenvalues of the nxn orthogonal projection matrix A onto V as 0 and 1 with respective eigenspaces V_perp. and V, and dimensions of ...
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0answers
16 views

Find symmetry along line

Let's consider line $L:(1,0,3)+t(2,2,1)$ in $\mathbb{R^3}$ find formula for symmetry along line $L$ I'd be greatful if anyone can describe the method how to approach this
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2answers
32 views

Linear Algebra - Finding row space and column space

My question is as follows: Find a basis for the row space and a basis for the column space by first reducing the matrix to row echelon form: $$ A =\left[\begin{array}{rrr} 5 & 9 & 3 \...
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1answer
244 views

Building a Diet Using Linear Algebra

The Question Suppose a diet calls for 7 units of fats, 9 units of protein and 16 units of carbohydrates for the main meal. Suppose the dieter has 3 possible types of food to satisfy this requirement: ...
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1answer
44 views

How do I show there exists a real matrix T such that

We have the following matrices : $$A=\begin{bmatrix}2 & 1 &-1\\1 & 2 & -1 \\-1&-1&4\end{bmatrix}$$ $$B=\begin{bmatrix}1 & 0 &-4\\0 & 5 & 4 \\-4&4&3 \end{...
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4answers
67 views

Double summation counting

I have the following: $$T_n=\sum_{j=1}^n\left(\sum_{k=1}^jk\right)$$ I know that the counting of numbers $1$ to $n$ can be expressed as $$\frac{(n+1)n}{2},$$ which leaves me with $$T_n=\sum_{j=0}^n\...
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1answer
19 views

Hil 2-cipher with 26 letter alphabet

A Hil 2-cipher with a 26-letter alphabet $A=1, B=2, \dots, Y=25, Z=0$ has enciphering matrix $A = \begin{bmatrix}19 & 13 \\ 6 & 3\end{bmatrix}$ Questions Verify that $A$ is suitable ...
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43 views

How to find he eigenvalues and eigenvectors of $L$?

Let $L:\Bbb R^3\rightarrow \Bbb R^3$ be defined by $L[a_1,a_2,a_3]=[-a_2, a_1+a_2, a_1-a_3]$. Using natural basis for $\Bbb R^3$, find the eigenvalues and eigenvectors of $L$. How do I represent ...
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57 views

vector-matrix multiplication upperbound

Let $\mathbf{p} = (p_i)_{i=1}^n$ with $p_i > 0$ be a ($n\times 1$)-column vector and consider a symmetric ($n\times n$)-matrix $\mathbf{A} = [a_{ij}]$ with $a_{ii} = 0$ and $a_{ij} \in [0,1]$ for $...
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Is there a simple relationship between the eigenvalues of a graph and a transition matrix?

Let $A$ be a adjacency matrix defining a graph, in which $A_{ij}=1$ if there is an edge between $i$ and $j$ and $A_{ij}=0$ otherwise. Let $P_{ij}=\frac{A_{ij}}{k_i}$ in which $k_i=\text{sum of }A_{ij}$...
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45 views

If $A$ is normal, show that $A$ and $A*$ are simultaneously diagonalizable.

If $A$ is a normal matrix, show that $A$ and $A^*$ (conjugate transpose) are simultenously diagonalizable over the field of complex numbers. I know: $AA^* = A^*A$ If $A$ and $B$ are simultaneosly ...
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Explicit matrices

Calculate explicit the matrices, which arise when $A=\begin{bmatrix}a&b\\c&d\end{bmatrix} \in Mat_2(C)$ is used in the following three polynomials: $f(T)=T^2 +1, \ g(T)=(T-i)(T+i), \ h(T)=\...
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2answers
49 views

Linear álgebra. Question about symmetric matrix

Prove that if $(Au)\cdot v = u\cdot (Av)$ for every $u$ and $v$ then $A$ is symmetric. I tried using a $2\times2$, but didn't help.
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1answer
59 views

What is the name of the measurement along a 4th dimensional axis?

Given that measurement along the X, Y and Z axes correspond to the terms "width", "height", and "depth", is there an accepted term for spatial measurement along the W axis when dealing in four ...
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88 views

Show that $F_2^4$ is a union of three proper subspaces

I'm just a bit confused about getting my head around this. I have seen proofs that say that a union of subspaces is only a subspace iff at least one subspace contains all the others, so, I'm not sure ...
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2answers
70 views

How to prove: for every two complementary subspaces there exists a projector

In Trefethen and Bau's book, Computational Linear Algebra, in the Projections chapter I've come across the following statement: Let $S_1$ and $S_2$be two subspaces of $\mathbb{C}^m$ such that $S_1 ...
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6answers
71 views

Prove this polynomial falls within $\mathbb R[x]$

[ The problem below is from Yao Musheng (姚慕生), Wu Quanshui (吴泉水), Advanced Algebra (高等代数学) Ed $2$, Fudan University Press, page $207$. ] Suppose $f(x)\in \mathbb C[x]$. If $\forall c\in \mathbb R$,...
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Let $T$ be a linear transformation on a vector space $V$ ($\textrm{dim}\ V = n$). If $\textrm{rank}\ (T^2) = n$, is $T$ invertible?

For a linear transformation $T$ on a finite dimensional vector space $V$ ($\textrm{dim}\ V = n$). If $\textrm{rank}\ (T^2) = n$, is $T$ invertible? Also, is it guaranteed to have an eigenvalue?
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1answer
15 views

why ${C^n} = C({A^*}) \oplus N(A)$?

Let $A \in {M_n}(C)$.Is this true that ${C^n} = C({A^*}) \oplus N(A)$?(where $C(B)$ is column space and $N(B)$ is null space of $B$)
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1answer
71 views

Eigenvalue and eigenvector implies that matrix $A$ satisfies $a_{ik}=a_{ij}a_{jk}$

Consider the following theorem (2.2). The author says the 'if' part is obvious so the proof was not given. Theorem 2.1 above says that every positive $n \times n$ matrix whose elements satisfy the ...
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1answer
50 views

Krein-Rutman for cones with empty interior

My question concerns the following theorem (a finite-dimensional version of Krein-Rutman): Let $V$ be a finite dimensional real normed space and $C \subseteq V$ a closed cone (i.e. a convex subset ...
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1answer
41 views

Mixture of mixtures.

I am formulating this problem for work, so it is important that I get it right. As of right now I am only considering the case where the number of chemicals is equal to the number of pre-made ...
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2answers
467 views

Determinant of the matrix

Suppose we have $n \times n$ matrix $A$ with $a_{i,j}={\rm gcd}(i,j)$. What is the determinant of $A$? Any ideas, folks?
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1answer
31 views

Row-reduced echelon

I apologize in advance if this is a very easy exercise. Let $k$ be a field and let $A$ be an $n \times n$ matrix over $k$ such that $A$ is a row-reduced echelon matrix without zero rows. Prove that $A$...
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2answers
163 views

Kernel of bilinear form

It is written in book, I read: kernel of bilinear form is space consisting of vectors $y$, such: $$Ker(\alpha)=\{y\in V:\alpha(x,y)=0,\ \forall x\in V\}$$ Nice I get it, but then it is said, that ...
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1answer
94 views

How to Formulate this Linear Algebra Fact in a Coordinate Free way?

There is a result result given in the last paragraph of pg 15 in Hoffman And Kunze's Linear algebra (2nd Edition) which essentially says that THEOREM. Let $F_1$ be a subfield of a field $F$. If ...
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39 views

Surd-like trinomials form a field

This is a problem from Artin's book "Algebra". In the fifth miscellaneous problem of the chapter "Vector spaces", he has asked to prove that: If $\alpha$ is a cube root of $2$, then the real numbers $...
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1answer
23 views

Difficulty understanding the method of undetermined coefficients.

I have to find the particular solution for this equation: $$y'' - 4y' + y = t*e^t + t$$ My initial thought was to use linearity and find the particular solution for both $t*e^t$ and $t$ and then ...
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Two vectors $f^r, f^{r+1} \in End_K(V)$ and f has maximum 2 eigenvalues

Let V be a finite-dimensional K-vector space, and $f: V \rightarrow V$ is an endomorphism. Suppose for $r \ge 0$ are the two vectors $f^r, f^{r+1} \in End_K(V)$ linearly dependent. Show that f has ...
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0answers
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principal components of mutual covariance marix

Principal component analysis basically takes the longest eingenvectors $\vec u_i$ of covariacne matrix $C_{ij}= \sum_k X_{ki} X_{kj} $ where $X_{ki}$ is $i$-th component of $k$-th data sample from ...
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1answer
31 views

what can I say about the matrix $A$?

Let A be real square matrix of order $n \geq 6$. if $a < 0$ is eigenvalue of $A^2$, then A is symmetric. if A has no real eigenvalues, then its invariant subspaces are only $\mathbb{R}^n$ ...
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1answer
73 views

A matrix with integers entries and complex solution. [duplicate]

Let $A$ be a $3\times 4$ and $b$ be $3\times 1$ matrix with integers entries. Suppose that the system $AX=b$ has a complex solution. Prove that the system $AX=b$ has a rational solution and set of ...
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1answer
30 views

find the rank of the given matix

Let $A$ be a matrix of order $n$ where $A=(A_{ij})$ and $A_{ij} = \min\{i,j\}$. Find rank of $A$. I am trying to find rows which generate the whole matrix.
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3answers
57 views

Let A be a $n\times n$ non-singular symmetric matrix with entries in $(0,\infty)$.

Let $A$ be a $n\times n$ non-singular symmetric matrix with entries in $(0,\infty)$. Then we can conclude that (a) $|A| > 0 (|A|$ denotes the determinant of A). (b) $A$ is a positive definite ...
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1answer
242 views

Is Summation a linear operator

Say I have the following recursive function, where superscript does not denote a power, but instead denotes a point in time. And I wonder if it's linear for the points in space (denoted by subscript i)...
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1answer
15 views

What is the basis of T?

Let $T: \Bbb{R^3} \to \Bbb{R^3}$ be a linear transformation: $T(a,b,c) = (a+b+c,0,0)$ Find a basis for $\text{Nullspace}(T)$. I found that the basis spans $(-1,1,0)$ and $(-1,0,-1)$, is this true ? ...
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1answer
52 views

How do I prove that if $S$ is onto then $T$ cannot be one-to-one?

Let $T: \Bbb{R}^3 \to \Bbb{R}^4$ and $S: \Bbb{R}^4 \to \Bbb{R}^2$ be two linear transformations such that the composition $S\circ T=0$. How can I show that if $S$ is onto, then $T$ cannot be one-to-...
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1answer
312 views

Suggest a follow up book to Axler's Linear Algebra Done Right?

So I know that a similar question has probably been asked about alternatives or compliments to this book, but I think my situation is different enough to warrant slightly different advice. So I've ...
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Existing complete function space allowing discontinuity .

This is a question which came to me due to several previous question: sorry for the all previous links necessary to look to get the question. The latest question is in the link: Convergence on Norm ...
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Need help finding the projection of a vector onto a subspace.

(a) Find the projection of the vector $\vec b=(2,1,0,1)$ onto the subspace $V$ consisting of all vectors of the form $(x_1,x_2,x_3,x_4)$ such that $x_1+x_2+x_3+x_4=0$. (b) What is the distance from ...
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35 views

Simple transformation question

(a) Let $B=\{1,t,t^2\}$ be the standard basis for $P_2(\mathbb R)$. Compute $[T]^B_B$. Solution: Since $T(1)=1$, $T(t)=t+1$ and $T\left(t^2\right)=(t+1)^2$, we find $$[T]^B_B=\begin{pmatrix}1&...
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1answer
42 views

Rank of the given matrix [closed]

Let $A=(A_{ij})$ be a matrix of order $n$, where $A_{ij}= i+j$. Find the rank of $A$.
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1answer
90 views

System of equations from weighted Gaussian Quadrature

I've been working on a weighted Gaussian Quadrature problem for a Numerical Analysis class and have been having the hardest time. The problem boils down to solving a system of four equations: $$ \...
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2answers
76 views

What happens to a system of difference equations when A is non-diagonalizable?

Suppose I have a system of linear difference equations $$ \mathbf{x}_{n+1} = A \mathbf{x}_n \>.$$ If $A$ is diagonalizable, then it can be shown that the system asymptotically approaches $\vec{0}$...
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2answers
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special relation between the expression of a matrix by block and its rank

Given two matrices $A$ and $C$ of order $n\times n$ and $m\times n$ respectively. We define the following matrix by block: $$ D=\left( \begin{matrix} C\\ CA\\ :\\ :\\ CA^{q-1} \end{matrix}\right) $$ ...
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33 views

Determining linear independence

Determine whether the vectors in $W$ are linearly independent. $W=\{(1,0,1),(3,4,5),(6, 5,1),(7,9, 2)\}$ Is there some way to show that these are linearly dependent? It seems obvious because there ...
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3answers
41 views

Simple inner product question.

Is there ever a case where ||x||$^2$ will not equal $<x,x>$? I don't understand why ||x||$^2$ is used sometimes.
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109 views

Convergence on Norm vector space.

I am not sure if this question make sense mathematically, so please bear with my ignorance. This is an extension to the question in the link: Is complete metric space is required? It seems in many ...
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1answer
40 views

Eigenvalue Sums

Problem 1: Let A and B be any 2 × 2 matrices. Compute the sum of the the eigenvalues of the matrix: C = AB − BA Hint: your answer should not depend on A or B.