Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

learn more… | top users | synonyms

1
vote
2answers
48 views

Find $A^{-1}$(W) of linear manifold W

Given linear map $A:\mathbb{R}^2\to \mathbb{R}^4$ defined as $$A = \begin{pmatrix} 1 & 1 \\ 1 & -1 \\ 0 & 2 \\ 3 & 1 \end{pmatrix}$$ and linear manifold $ W \subset ...
5
votes
0answers
932 views

Possible Jordan Canonical Forms Given Minimal Polynomial

I was supposed to find all possible Jordan canonical forms of a $5\times 5$ complex matrix with minimal polynomial $(x-2)^2(x-1)$ on a qualifying exam last semester. I took the polynomial to mean ...
0
votes
1answer
79 views

Finding polynomials p(X) and q(X) subject to conditions on a vector space V

$V$ is the vector space $\mathbb{R}_{2}\left [ X \right ]$ of polynomials of degree at most 2 in $X$ with real coefficients. $\left \langle f(X),g(X) \right \rangle$ is the inner product ...
0
votes
1answer
112 views

Minimization to Maximization doubt in SVM

I came across a lecture on Support Vector Machines and in the lecture they converted a maximization problem into a minimization problem. I am wondering how it was done... $ Max \frac {1}{||x||} $ ...
1
vote
2answers
68 views

Some questions about scalar product

Definition of scalar product: if $g:V \times V\to\Bbb{R}$ is nondegenerate symmetric bilinear form, $g$ is a scalar product on $V$ vector space. Here is my question If $g$ scalar product is indefinite ...
3
votes
1answer
82 views

Check If a point on a circle is left or right of a point

What is the best way to determine if a point on a circle is to the left or to the right of another point on that same circle?
1
vote
1answer
505 views

Complementary Subspaces

In $\mathbb{R}^4$ we have a subspace of all 4-tuples $(a,b,b,c)$ where the second and third component are equal. Is there a complementary subspace that completes it to $\mathbb{R}^4$? (we need ...
2
votes
2answers
139 views

prove that a linear map is injective - $T(X) = X + 2X^T$

I have the following linear map: $$T: \operatorname{Mat}_{n\times n}(\mathbb{R}) \to \operatorname{Mat}_{n\times n}(\mathbb{R})\;,$$ $$T(X) = X+2X^T\;.$$ I have to prove that it is injective ...
0
votes
1answer
140 views

A question about symmetric bilinear forms

If b is an indefinite symmetric bilinear form is it nondegenerate? And conversely if b is nondegenerate is it positive/negative definite or indefinite? How can i start to prove this? Note:Edited and ...
6
votes
3answers
179 views

Simplicity of eigenvalue

I have a matrix $A$ and I introduce $(I+A)^{m},$ where $I$ is the identity matrix of same order with $A$ and $m$ is a positive integer. I want to show that if $(1+ \lambda )^m$ is a simple eigenvalue ...
2
votes
1answer
99 views

an $ N \times N $ matrix with positive integral entries

A magic square of order $N$ is an $ N \times N $ matrix with positive integral entries such that the elements of every row, every column and the two diagonals all add up to the same number. If a magic ...
2
votes
4answers
4k views

Find intersection of two 3D lines

I have two lines $(5,5,4) (10,10,6)$ and $(5,5,5) (10,10,3)$ with same $x$, $y$ and difference in $z$ values. Please some body tell me how can I find the intersection of these lines. EDIT: By using ...
2
votes
1answer
35 views

nondegeneracy of quadratic space

For a nondegenerate quadratic space $(V,B)$ and $W \subset V$.Prove the followings are equal (i) $W\bigcap W^\perp =0$. (ii) $W$ is nondegenerate. (iii)$W^\perp $ is nondegenerate. I tried but i ...
1
vote
1answer
140 views

Separation Theorem in Euclidean Space.

I want to show the following: Let $A,B \subseteq \mathbb{R}^n$ disjoint, nonempty, closed and convex sets. Then there exists a $h \in \mathbb{R}^n$, such that $A$ and $B$ gets separated in the ...
1
vote
2answers
82 views

Showing that a linear map is an isomorphism

Let $V$ a vector space and $W$ be its linear subspace. Give an example of a linear map that satisfies $\mathrm{im}(f)=W$ and $\ker(f) \oplus \mathrm{im}(f)=V$, but $f^2 \neq f$. Would $f(v)=2v$ be ...
1
vote
3answers
84 views

How to solve this equation in two variables

My question is: How to solve this equation: $ax²+by²+cxy=0$ with respect to $x$ and $y$ in the same time. Here $a,b,c$ are real constants.
2
votes
3answers
1k views

If both roots of the Quadratic Equation are similar then prove that

If both roots of the equation $(a-b)x^2+(b-c)x+(c-a)=0$ are equal, prove that $2a=b+c$. Things should be known: Roots of a Quadratic Equations can be identified by: The roots can be ...
1
vote
0answers
110 views

What is the condition for a polynomial to be factorizable in linear real factors?

I have a polynomial $p_a(x,y)= x^2F(a)+y^2G(a)-xH(a)-I(a)$ where $F(a)$, $G(a)$, $H(a)$ and $I(a)$ some real fuctions of $a$ are. Which conditions must satisfy $a$ so that I can factorize the ...
2
votes
0answers
53 views

Solution to pertubed linear system

Suppose one has the following system of linear equations $$(A + \Delta A) x = b$$ where $A$ and $\Delta A$ are large sparse matrices and $\Delta A$ is "small" compared to $A$, furthermore vector $x$ ...
0
votes
1answer
42 views

how to prove $\|(A^HA)^k\| =||A||^{2k}$ using singular value decomposition

how to prove $\|(A^HA)^k\| =||A||^{2k}$ using singular value decomosition. $A^H$ is a hermitian matrix. $A$ element of $C^{p\times q}$, for every positive integer $k$.
1
vote
2answers
40 views

Linear regression method help?

Let's suppose we have a function $Y=A\cdot t^B$ and the values for $Y$ are $30,60,90,120,150$ and the values for $t$ are respectively $0.974, 1.331, 1.718, 1.971, 2.356$. Can you find $A$ and $B$ ...
3
votes
3answers
257 views

Symmetric positive definite matrix inequality

Hi could you help me with the following: Show that for a symmetric positive definite matrix $B$, $$b_{ij} + b_{jk} + b_{ki} \leqslant b_{ii} + b_{jj} + b_{kk}$$ holds for any $1 \leqslant i,j,k ...
2
votes
1answer
381 views

The form of 2 by 2 unitary matrices

I've been working through "Groups and Symmetry" (Armstrong) and came across this problem in chapter 9 which I can't figure out. Any hints/help would be greatly appreciated! Show that every $2\times2$ ...
1
vote
2answers
104 views

Diagonalizability of a certain matrix from a given field.

This is a question from Serre's Exercise book in Matrix theory. I don't even know how to start. Any help would be appreciated. Assume that the characteristic of the field $k$ is not equal to 2. ...
4
votes
1answer
248 views

Intuitive interpretation of the adjacency matrix as a linear operator.

Naturally we can describe graphs via tables of "yes there is an edge" or "no there is not" between each pair of vertices, so the definition of an adjacency matrix is easily understood. Thinking of ...
0
votes
1answer
81 views

iteration convergence bounds with norm less than 1

Let $x_{k+1} = Bx_k + c$ where $B$ is $n \times n$ matrix $c$ is a vector. Assume $\|B\| \le \beta <1$ $\|x_k - x_{k-1}\| \le \varepsilon$ for some $k$ Show that $\| x - x_k\| \le ...
3
votes
1answer
57 views

Eigenvalues of matrix - only $1$ or $0$

Let $A$ be a matrix so $A=A^2$. I need to show that the eigenvalues of $A$ are only $1$ or $0$. I tried some ways but none of them help.
1
vote
1answer
49 views

kernel of linear combination

Let $T,S: V\to V$ be 2 linear transformations, and $\ker(T)=\{0\}$. I need to prove why $\ker(T\circ S)=\ker(S)$. I have no idea how to prove it.
5
votes
4answers
436 views

Vector space of polynomials over $\mathbb{R}$ with degree $\leqslant n-1$

Let $P \in \mathbb{R}_{n-1}[X]$ be a polynomial of degree $n-1 \geqslant 0$. Let $\mathbb{R}_{n-1}[X]$ be the vector space of polynomials with degree $\leqslant n-1$ over $\mathbb{R}$. Show ...
4
votes
2answers
271 views

Asymptotics of system of linear equations

I have a system of linear equations as follows. $$M(p) = 1+\frac{n-p-1}{n}M(n-1) + \frac{2}{n} N(p-1) + \frac{p-1}{n}M(p-1)$$ $$N(p) = 1+\frac{n-p-1}{n}M(n-1) + \frac{p}{n}N(p-1)$$ $$M(1) = ...
5
votes
1answer
135 views

real eigenvalue

Let matrix $A$ be $$\begin{bmatrix} -5& 1& 0& 0\\ a &2& 1 &0\\ 0& 1 &1 &1\\ 0 &0&1& 0 \end{bmatrix}$$ where $a$ is a constant between 1 ...
0
votes
2answers
127 views

Injective homomorphism $(\mathbb{Z}_{12}, +) \to (\mathbb{Z}_{18}, +)$

$f:G \rightarrow H$ is a group homomorphism $g\in G$ is an element of order $n$. (a) Prove that $f (g)$ is a final order and that the order of the element $f (g)$ parts n. (b) What is the order of ...
0
votes
1answer
429 views

norm less than 1 matrix theory

Let $A$ be any $n \times n$ matrix and $\| \cdot \|$ be the matrix norm induced by vector norm on $\mathbb{R}^n$ (Euclidean n-dimensional space). If $\|I - A\| < 1$, then show that $A$ is ...
1
vote
2answers
104 views

Matrix norm less than $1$ iteration

Is the following true always for a matrix norm $$\lVert AB\rVert \leqslant \lVert A\rVert \cdot \lVert B\rVert \text{ ?}$$ Related to this given $r$ is positive constant, $H$ is symmetric positive ...
3
votes
4answers
73 views

Determining vector equations

Let $A\in \Bbb R^{n\times n}$ be a matrix such that $\mathrm{rank}(A) = n-1$ and consider the equation $$ Ax = 0. $$ Clearly, its solutions span a $1$-dimensional space, thus an additional ...
1
vote
1answer
265 views

Largest eigenvalue, semidefinite programming

The problem is t minimize the largest eigenvalue of a function of x. objective: $$ min \ \ \ \lambda_{max}(A(x))$$ where $$A(x) = A_0+x_1A_1+x_2A_2+...x_nA_n$$ and all $A$ is positive semidefinite. ...
7
votes
1answer
206 views

Isomorphism of rings implies isomorphism of vector spaces?

Let $A$ and $B$ be isomorphic unitary rings. Suppose that both of them admit a structure of (maybe finite dimensional) vector space over some field $k$. I would like to know if then $A$ and $B$ are ...
2
votes
1answer
109 views

Is this subspace linear or affine?

I have several subspaces where I have to determine their dimension and whether they are affine or linear? These are my answers- are they correct? Thanks for help! a) $X = \{ x \in R^n | a^Tx = 0 \}, ...
1
vote
0answers
73 views

Time complexity of Gaussian elimination over polynomial ring

I have a $t \times l$-polynomial matrix $A$ over $\mathbb{F}_q[x]$. The entries of $A$ are of degree $\le m$. I want to reduce $A$ to upper-triangular form by Gaussian elimination in case of using the ...
9
votes
8answers
2k views

Linear independence of $\sin(x)$ and $\cos(x)$

In the vector space of $f:\mathbb R \to \mathbb R$, how do I prove that functions $\sin(x)$ and $\cos(x)$ are linearly independent. By def., two elements of a vector space are linearly independent if ...
0
votes
3answers
176 views

Linear algebra proofs - traces, symmetricity and inversion

I have a few proofs I need some help with. a) Prove that $AB-BA = I$ does not have any solutions for any $A,B$. All matrices are regular. I based my proof on matrix traces. $tr(AB) = tr(BA)$. Since ...
2
votes
1answer
58 views

Are these systems of equations linear?

Are these sets of equations linear? What is the number of variables and equations in each system? Please correct me if my answer is wrong: a) $Ax = b, x \in R^n$ - yes, classic system of linear ...
3
votes
1answer
111 views

Solving ill posed linear equations

Given a set of linear equations $AX=B$, say $A$ is an ill posed matrix (has a few singular values equal or very close to zero), which numerical algorithm (conjugate gradient, least squares or steepest ...
1
vote
2answers
225 views

Eigenvalues of the matrix $(I-P)$

Let $P$ be a strictly positive $n\times n$ stochastic matrix. I hope to find out the stability of a system characterized by the matrix $(I-P)$. So I'm interested in knowing under what condition on ...
0
votes
1answer
152 views

Spectrum of doubly stochastic matrices

Let $M$ be a doubly stochastic matrix in which every entry is strictly positive. Prove that for any eigenvalue $\lambda$ we have $\lambda \neq 1 \Longrightarrow |\lambda|< 1$ and the geometric and ...
0
votes
2answers
94 views

Suppose R is a commutative ring, $A \in R_n$, and the homomorphism $f: R^n \to R^n$ defined by $f(b) = Ab$ is surjective. Show $f$ is an isomorphism.

How would I go about showing this? Suppose $R$ is a commutative ring, $A \in R_n$, and the homomorphism $f: R^n \to R^n$ defined by $f(b) = Ab$ is surjective. Show $f$ is an isomorphism. (Edit: ...
5
votes
3answers
376 views

prove that $\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$

If $A$ is a $m \times n$ matrix and $B$ a $n \times k$ matrix, prove that $$\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$$ Also show when equality occurs.
1
vote
0answers
37 views

How to prove the existence of a factorization with the given precision for ID?

Let us decompose matrix $A \in \mathbb{R}^{m\times n}$ as the multiplication of matrices $B \in \mathbb{R}^{m\times k}$ and $P \in \mathbb{R}^{k\times n}$ where some subset of the columns of $P$ make ...
3
votes
1answer
66 views

How to be sure that the $k$th largest singular value is at least 1 of a matrix containing a k-by-k identity

In section 8.4 of the report of ID software, it says that the $k$th largest singular value of a $k \times n$ matrix $P$ is at least 1 if some subset of its columns makes up a $k\times k$ identity. I ...
4
votes
1answer
310 views

Largest eigenvalue of a $A^T A$ matrix?

I have a large real matrix A of size $40K\times 400K$, is there an efficient way to calculate the largest eigenvalue of $A^T A$ (size $400K\times 400K$)? Thanks.