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 ...

learn more… | top users | synonyms (1)

0
votes
1answer
45 views

Where am I going wrong with finding eigenvectors?

Simple example, but I am having the same issues with all of the problems I attempt. $$A= \left[\begin{array}{rrr|r} 6 & 3 \\ 2 & 7\\ \end{array}\right] $$ I get eigenvalues 4 ...
2
votes
0answers
37 views

Find a matrix to represent the mapping of a factor module

I have a problem from my past paper I can't figure the logic to, even after seeing the answers. The question goes 【Q】Let $V=\mathbb{R}[X]_{<4}$ be the vector space of real polynomials of degree ...
1
vote
1answer
132 views

Using SVD to approximate matrix-vector multiplication?

Given some matrix A, is it possible to use Singular Value Decomposition to approximate Ax for some vector x within some error bound? According to Efficient low rank matrix-vector multiplication, it ...
3
votes
1answer
158 views

Definition of Distinct eigenvalue clarification?

I'm solving a problem where I am given the eigenvalues of a matrix $A$ and need to solve for the determinant of $A$. I know that if my matrix is diagonalizable I can find the determinant of $A$ by ...
0
votes
1answer
42 views

Why is this valid $tr(VDV^{-1}) = tr(VV^{-1}D)$?

Given a diagonalizable matrix A, such that this relation holds: $A = VDV^{-1} $, where D is a diagonal matrix. Now the following relation is given when taking the trace: $tr(VDV^{-1}) = tr(VV^{-1}D) =...
1
vote
0answers
32 views

Gauss-Jordan Algorithm underdetermined equation system

$Ax=b$ with $A$ more columns than rows. If I apply Gauß Jordan algorithm, I get a diagonal matrix, where I can read off one solution. But don't I loose some solutions, because there are many ...
0
votes
1answer
47 views

Elementary Transformation

$Ax=b$ with $A$ integer Matrix and $b$ integer vector. Looking for solutions $x_i$ in $\mathbb{Z}$. So if we multiply by elementary transformation matrices: (Add an integer multiply of one column/...
0
votes
1answer
89 views

Find the best approximation (in $L^2$ / mean-square sense) for $ln x$

The full statement of the problem is: Consider the set of two functions $\{1,x\}$ on the interval $x\in[0,1]$. Replace the second function by another one in $span\{1,x\}$ which turns the pair into an ...
1
vote
1answer
42 views

Similarly Commuting Matrices

I was wondering, if $D$ and $A$ are similar matrices, over $\mathbb{R,C}$, that is $D=S^{-1}AS$ and $DC=CD$, for some $C$, must $A$ commute with $C$? For some reason, this one is slipping from me. I ...
0
votes
4answers
356 views

Suppose T is a linear transformation such that T(1,1,1)=(0,1,2), T(1,0,1)=(1,1,1), and T(0,0,1)=(1,2,3).

What is T(x,y,z) What I tried: T(x,y,z)= $$ \begin{pmatrix} 1&1&1|0&1&2\\ 1&0&1|1&1&1\\ 0&0&1|1&2&3\\ \end{pmatrix} $$ Which reduces to \begin{...
0
votes
2answers
62 views

Well-defined and Equivalence relations

I am wondering why the following is well-defined... The definition of well-defined is given as; $g:(X/\sim) \to Z$ is well-defined if a mapping $f:X \to Z$ can be found where $f$ has the property $x \...
1
vote
1answer
25 views

Use separation of variables to find a solution $u= u(x,t)$…

So I get up to the last paragraph of the solution. I can get the bases of the solution, but beyond that, I'm really confused as to what they did. Any help would be appreciated!
-2
votes
1answer
38 views

Have a question about Linear Transformations

Explain why there cannot be a linear transformation T: $R^2$ --> $R^2$ for which T(1,1)=(2,3) and T(3,3)=(1,4). I have no clue how to start this problem. Wouldn't $T(4,4)$=$4T(1,1)$=$4(2,3)$=$(8,12)$...
0
votes
1answer
52 views

If the entries of an invertible matrix N are between -1 and 1, is its operator norm less than 1?

For Euclidean norm. If so, why? If not, might $(I-N)^{-1}$ exist some other way? This spins-off from here.
-2
votes
4answers
95 views

Prove that $ A = - A^{\top} $ and $ \text{rank}(A) \leq 1 $ imply $ A = \mathbf{0} $. [closed]

Let $ A \in {\text{M}_{n}}(\Bbb{R}) $, and suppose that we have the following: $ A = - A^{\top} $. $ \text{rank}(A) \leq 1 $. Why then is $ A = \mathbf{0} $? Thanks!
0
votes
0answers
35 views

linear algebra bound

I have a problem in course project? I have two positive definite $n\times n$ matrices, $A$ and $B$. I want to find the bound of singular values of a product of these matrices ? these matrices are ...
0
votes
0answers
31 views

Find all $(x,y,z)\in \Bbb Z^3$ so that $ x^y=x(\text{mod }z)$

I got this problem for Integer Triplets I tried this way : $ x^y=x(\text{mod }z)$ which implies $ x^y-x=0(\text{mod }z)$ and then i got stuck, What should I do? is their a better way
0
votes
1answer
125 views

Theorem** on page 288 from Geometry of algebraic curves by Arbarello, Cornalba, Griffiths, Harris

1Let $V$ a $2n$ complex vector space and take on $V$ a quadratic form. Now define $$ \Sigma=\{\Lambda:Q(\Lambda,\Lambda)\equiv0 \} \subset Gr(n,2n)$$ where $\Lambda$ is a maximal subspace i.e it is ...
4
votes
2answers
110 views

How to show $f(x)$ has no root within $\Bbb Q$

A polynomial problem from my old algebra textbook: $f(x)\in\Bbb Z[x]$ with leading coefficient $1$, $\deg f(x)\ge 1$, and both $f(0)$ and $f(1)$ are odd numbers, prove: $f(x)$ has no root within ...
1
vote
1answer
43 views

What seems to be the minors of the Adjugate matrix $\text{adj}(A)$ of a square matrix $A$?

It is by definition that entries of the adjugate matrix $\text{adj}(A)$ are the corresponding $(n-1)$-minors of $A$ (up to a sign). What can we say about the $k$-minor of $\text{adj}(A)$ in relation ...
1
vote
1answer
49 views

Eigenvector of a $C^n$ class matrix

Let $A$ be the following matrix function: $\Bbb{R} \to \Bbb{R}^{a \times (a+1)}$ $t \mapsto A(t)$ Let us suppose that $A$ is $C^{\infty}$, meaning that all of $A$ coefficients are $C^{\infty}$. Let ...
2
votes
1answer
59 views

Theorem 3.1 in Erwin Kreyszig's “Introductory Functional Analysis With Applications”: Is the notion of convex set valid in complex vector spaces?

Is the notion of convex sets valid and meaningful for complex vector spaces? Or, do we need to restrict ourselves to real vector spaces and normed spaces when we talk about convex sets? The ...
0
votes
1answer
45 views

$A$ is singular and normal matrix, what must be its characteristic polynomial?

Let $A$ be a $5\times5$ real singular matrix which is normal. If $1-2i$ is an eigenvalue of $A$ and $2+i$ is an eigenvalue of $A^*$ (conjugate transpose), what must be its characteristic polynomial? ...
0
votes
1answer
18 views

Do matrices have average and fluctuations?

Given a set of numbers, one can calculate the average of those numbers and the fluctuation (variance) over the average. E.g,, $\langle A \rangle=\frac{1}{N}\sum_{i=0}^N A_i$ and $(\delta A)^2 = \...
1
vote
1answer
46 views

Field extensions and algebraic elements

Can somebody explain why taking beta gives $K(\beta)$ as a subspace of $K(\alpha)$?
3
votes
2answers
122 views

Naturality in linear algebra

Question. How can we formalize these intuitions about predicates on matrices? Let $P$ denote a predicate on matrices, so that $P(A)$ is true for some choices of matrix $A$ and false for all others. ...
4
votes
1answer
89 views

Linear decomposition of positive semi-definite matrices

It is true that the vector space of $n\times n$ Hermitian matrices is an $n^2-$dimensional real vector space and that one can find a basis for this space consisting exclusively of positive semi-...
1
vote
2answers
81 views

Compute the limit of a matrix

We need to compute the limit of a sequence as $x \rightarrow \infty$ Using matrix-matrix multiplication we can define power as $A^p=A*\cdots*A$, $p$ times. We need to compute the limit of $A^p$ as $...
0
votes
0answers
34 views

Least square matrix form will fail, if the inverse property not satisfied?

In the matrix form of least squares , the inverse of ( X transpose X ) we are calculating . So, what if that matrix does not posses inverse properties. I mean what if it is not invertible ? Sorry if ...
3
votes
2answers
98 views

Under what conditions does $(I-N)^{-1}$ exist?

Given an nxn matrix N and $I=I_n$, under what conditions does $(I-N)^{-1}$ exist? On one hand $(I-N)(I + N + N^2 + ...) = (I + N + N^2 + ...) - (N + N^2 + ...) = I?$ On the other hand, $(I-N)(I + N +...
-5
votes
1answer
91 views

What is the least positive integer $m$ such that $\text{rank} A^m=\text{rank} A^{m+1}$? [closed]

Let $A$ be a complex square matrix. What is the least positive integer $m$ such that $\text{rank} A^m=\text{rank} A^{m+1}$? Express $m$ in terms of some quantities associated with $A$.
1
vote
3answers
30 views

If $T : V \to k$ is not the zero map, there is $v \in V$ such that $T(v) = 1$.

If $V$ is a $k$-vector space and $T : V \to k$ is a homomorphism (linear map) that is not the zero map, is it true that some element of $V$ is mapped to $1$ of $k$? Of course $k$ is a 1-dimensional $k$...
1
vote
0answers
38 views

Need more insight on a formula

Following is a part of a programming contest problem. Given $C_{1},m,n,o,x,y,z,c,d,K,J$ are positive integers $ C_{i} = \left\{ \begin{array}{l l} (m*C_{i-1}^2 + n*C_{i-1} + o) \bmod J & \...
0
votes
1answer
22 views

Condition on coefficients of a linear equation

We express a linear equation in two variables as: $a x + b y + c = 0$, where $a$ and $b$ both are not simultaneously equal to zero. That is, if $a\neq 0, b$ can be zero and vice versa but not $a=b=0$....
1
vote
4answers
78 views

Difficulty in understanding a step in a definition in the book Walter Rudin

In the book Principles of mathematical analysis by Walter Rudin,He writes: "For $ A\in L(\Bbb R^n,\Bbb R^m)$, define the norm $||A||$ of $A$ to be sup of all numbers $|Ax|$, where $x$ ranges over all ...
2
votes
1answer
60 views

Let A be an $n\times m$ matrix and B be an $m\times n$ matrix such that AB is invertible. Then which of the following is/are always true?

Let A be an $n\times m$ matrix and B be an $m\times n$ matrix. For a square matrix D, let Tr(D) denote trace of D, |D| denote the determinant of D. Suppose that AB is invertible. Then which of the ...
1
vote
1answer
53 views

How do we find eigenvalues from given eigenvectors of a given matrix?

For instance let $$A=\begin{pmatrix} 3 & -1 & -1 \\ 2 & 1 &-2 \\ 0 & -1 & 2 \\ \end{pmatrix}$$ be a matrix and $$u_1=\begin{pmatrix} ...
1
vote
1answer
45 views

Determining whether or not the subset S is a subspace

So I'm reading my linear algebra textbook where it says: Theorem 4.2: Let $S = \text{span}\{u_{1}, u_{2}, \cdots, u_{n}\}$ be a subset of $\mathbb R^{n}$. Then $S$ is a subspace of $\mathbb R^{n}...
2
votes
2answers
681 views

Determining if subset S = [a b c] where a, b, c are ≥ 0 is a subspace?

Determine if the described set is a subspace. Assume a, b, and c are real numbers. The subset of R3 consisting of vectors of the form [a b c] , where a ≥ 0, b ≥ 0, and c ≥ 0. Here is my reasoning so ...
1
vote
1answer
22 views

Conventional notation: $\vec{a^T}\vec{a} = ||\vec{a}||$ or $\vec{a^T}\vec{a} = ||\vec{a}||^2$

Just want to make sure I grasp the conventional notation on this once and for all. If I recollect correctly the following should be correct, right?: $\vec{a}\cdot \vec{a} = \vec{a^T}\vec{a} = ||\vec{...
0
votes
1answer
19 views

Problem on direct sum

Consider subspaces of $\mathbb{R}^n$: $L_1, L_2, L_3$ with dimensions $n_1, n_2, n_3$, respectively. It's known that $n_1 + n_2 + n_3 = n$ and $L_1 + L_2 + L_3 = \mathbb{R}^n$. Prove that sum ...
1
vote
0answers
23 views

Problem about construction of linear map using the Fundamental Theorem of Linear Map.

I have some problem with the theorem: Linear maps can be constructed that take on arbitrary values on a basis. Specifically, given a basis $\{v_1, \dots, v_n\}$ of $V$ and any choice of vectors $...
1
vote
2answers
43 views

Real square matrix of order 7 has 6-dimensional invariant subspace

Let A be real square matrix of order 7, then A has 6-dimensional invariant subspace. How to prove it?
3
votes
1answer
65 views

If a matrix commutes with two others, must the other two commute?

I am super confused on how to get started on this problem. A starting hint would be great. I am given that $A \in M_n(\mathbb{C})$, and that for $B,C \in M_n(\mathbb{C})$, $AB=BA, CA=AC$ and that $A$ ...
1
vote
0answers
87 views

Atlas on the Grassmannian Variety

Let $G(k,n)$ the set of all $k$-dimensional sub-spaces of a vector complex space $V$ of dimension $n$. I know that it is possible to define the grassmannian as the quotient of $\chi(n,k)$ by $GL(k)$ ...
3
votes
2answers
33 views

Any (n-1) vectors from system X are linearly independent $\nRightarrow$ X is linearly independent

Let X denote system of n vectors from $\mathbb{R}^n$. Any (n-1) vectors from X are linearly independent. Claim: the system X is linearly independent. I need a counterexample, because the claim ...
0
votes
0answers
33 views

How accurate the solution of over-determined linear system of equation could be using least square method?

I have read the theory of least square method. It is used to minimize the Frobenius norm of equation residual vector. but I searched the internet and I did not find how to determine the actual value ...
72
votes
10answers
4k views

Why, intuitively, is the order reversed when taking the transpose of the product?

It is well known that for invertible matrices $A,B$ of the same size we have $$(AB)^{-1}=B^{-1}A^{-1} $$ and a nice way for me to remember this is the following sentence: The opposite of putting ...
2
votes
2answers
77 views

If the eigenvalues of a matrix $A$ are $0, 1$, is $A$ is projection?

It's easy to prove that a projection has $0$ and/or $1$ as its eigenvalues. My question is: If a matrix has only $0$ and $1$ as its eigenvalues, does that mean that this matrix is a projection? (...
0
votes
2answers
123 views

True or false: The non-pivot columns of a matrix are always linearly dependent.

True or false: The non-pivot columns of a matrix are always linearly dependent. This is false, I just don't really understand why. Thanks for any help!!