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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Is there a difference between linear map and linear transform?

Wikipedia page says there is no difference but when I see reference to a map it's domain and range are specified but in a transform it is not the case. Any suggestion how to view the both?
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Definition of a matrix by using coaction.

Let $V$ be a vector space with a basis $e_1, \ldots, e_n$. Let $C$ be a coalgebra and $V$ a $C$-comodule. Consider the coaction $\delta: V \to C \otimes V$ given by $\delta(e_i) = \sum_{j=1}^n c_{ij} ...
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How Many Days Until 100 Screens [on hold]

I earn 2 dollars each day from a screen costing 200 dollars.In how many days can I buy 100 screens if invest back all my earnings to buy new screens? To start with I have $200 in hand and no ...
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17 views

Stabilizers of Segre varieties

What, if anything, is known about maps in PGL(V) that preserve Segre varieties? I am specifically interested in linear maps preserving the Segre embeddings of $\mathbb{P}^{15} \times ...
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2answers
25 views

Linear transformations that are diagonalizable and nilpotent

Let $V$ be a vector space of finite dimension. Find all linear transformations $T:V\rightarrow V$ that are both diagonalizable and nilpotent. I was thinking $T=0$. But are there other such ...
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1answer
23 views

Clamp Distance Between Two Vectors

I was wondering if there is a formula to clamp distance between two Vectors. Let me elaborate. I have two Vectors, say, $V_1(x_1,y_1)$ and $V_2(x_2,y_2)$. I can find the distance '$d$' between them ...
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3answers
581 views

classification up to similarity of complex n-by-n matrices

Classify up to similarity all 3 x 3 complex matrices $A$ such that $A^n$ = $I$.
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34 views

rational eigenvalues of integer matrix are integral

Let $A=A^T$ a real symmetric matrix with integer entries. How do you prove that a rational eigenvalue of A is integral?
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37 views

Orthogonality lemma sine and cosine

I want to know how much is the integral $\int_{0}^{L}\sin(nx)\cos(mx)dx$ when $m=n$ and in the case when $m\neq n$. I know the orthogonality lemma for the other cases, but not for this one.
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20 views

Simple vector algebra question for perpendicular lines

I'm actually a little ashamed to ask this, Suppose A and B are not 0. Consider a line "l" whose cartesian equation is $Ax+ By + D = 0$. Suppose that $P_0 = (x_0,y_0)$ does not lie on "l". Show that ...
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31 views

Find global minimum of the function

I need to find the global minimum of the function $$f ( x) = \langle Ax,x \rangle + 2\langle b ,x\rangle+c$$ where $c \in \mathbb{R}$ is constant, $b \in \mathbb {R}^n$, and $A$ is a positive ...
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Non-unique factorization in $\mathbb{Z}[\sqrt{-5}]$

I want to show that the decomposition into irreducible factors in the ring $$\mathbb{Z}[\sqrt{-5}] = \{a + b\sqrt{-5}|\space a, b \in \mathbb{Z}\}$$ is not unique, except for the order of factors ...
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How to implement QR method for bidiagonal matrices?

My goal is to take the singular value decomposition of a (not necessarily square) matrix. I have a method to do bidiagonalization of a matrix, and I can chop the bottom rows of zeros. In order to find ...
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11 views

Isomorphism inbetween a factor ring and Cartesian product of factor rings [duplicate]

Let R be principal ideal ring and $a_1, ..., a_n \in R$, with $gcf(a_i, a_j) = 1$, for all $i, j \in \{1, ..., n\}$ (with $gcf$ being the greatest common factor of $a_i, a_j$). Show that the ...
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1answer
35 views

Are $Mx = 0$ and $M^TMx = 0$ equivalent? [on hold]

Let $M$ be a $k$ by $n$ matrix with integer elements and $k < n$ and let $x \in \{-1,0,1\}^n$. Is it true that $Mx = 0$ if and only if $M^TMx = 0$?
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Help in doing algebraic manipulations.

Let the homogeneous system of linear equations $px+y+z=0$, $x+qy+z=0$, $x+y+rz=0$ where $p,q,r$ are not equal to $1$, have a non zero solution, then the value of $\frac{1}{1-p} + \frac{1}{1-q} + ...
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Ax=b has a solution if and only if (b,y)=0 [on hold]

Prove that Ax=b has a solution if and only if (b,y)=0, for all vectors y satisfying A* y = 0, where A* is the adjoint of A, and A is singular.
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32 views

Why a form is positive only if its matrix in some ordered basis is a positive matrix?

I'm reading Hoffman's "Linear Algebra" Chapter 9 "Operators on Inner Product Spaces" and got lost at the positive property on (sesqui-linear) forms, operators and matrices. The confusing comes from ...
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1answer
37 views

Why should there be a 7-dimensional cross product in the context of exterior algebra?

The three-dimensional cross product can be viewed as the wedge product corresponding to the exterior power $\Lambda^2(\mathbb R^3)$. An explanation that I have come up with for the scarcity of cross ...
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3answers
40 views

Inner product: $(x,z)=(y,z)\implies x=y$?

We've talked about inner products in our last tutorial and couldn't really get answered the following questions: Let $(\cdot,\cdot)$ be any inner product. If $(x,z)=(y,z)$ for all $z$ of any given ...
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1answer
43 views

Isotropic cone of a quadratic form

Let $E$ be a finite dimensional vector space over $K$. Let q be a quadratic form on $E$. Let $C(q)$ denote the isotropic cone of $q$ (the isotropic cone of a quadratic form $q$ is the set of all ...
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17 views

Length of a projected line

If a line is of true length x and is inclined in angles a,b,c with respect to the xy,yz,zx planes respectively , then how can i find the length of the projected line in the xy , yz and zx planes ...
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23 views

Fast method for getting solution for underdetermined equation system

What is a fast and stable method for getting a solution for an underdetermined equation system which could be applied by a computer?
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53 views

About polynomials. If there are no two polynomials of the same degree in $S$, then $S$ is linearly independent.

The problem states as follows. Let $S$ be a set of non-zero polynomials over a field$ F$. If there are no two pplynomials of the same degree in $S$, then $S$ is linearly independent. I tried the ...
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range of $m$ such that the equation $|x^2-3x+2|=mx$ has 4 real answers.

Find range of $m$ such that the equation $|x^2-3x+2|=mx$ has 4 distinct real solutions $\alpha,\beta,\gamma,\delta$ To show how I got the wrong answers. From $|x^2-3x+2|=mx$ I got the two case ...
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20 views

identity operator, direct sums, and projections

Let W be finite-dimensional vector space. Let $P: W\to W$ be a projection. Let U = Range(P) and V=Ker(P) (a) show that P is the identity operator on U. I dont understand the problem ...
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11 views

linear transformation sequences

I'm working on this exercize about linear transformation: Let $E=\mathbb{R^N}$, $T:E \rightarrow E: \ (u_n)_{n\geq 0} \rightarrow (u_{n+1})_{n\geq 0}$ $S:E \rightarrow E: \ (u_n)_{n\geq 0} ...
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50 views

Is there a problem in assuming that a point is the same thing of a vector?

I've read Apostol's Calculus, in the section on analytic geometry. He says that he's going to use 'vector' and 'point' interchangeably. But in Beardon's Algebra and Geometry, he argues that there is ...
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1answer
27 views

Showing $T:K^n \to K^{n-1}$ is surjective

Hi everyone, I'm a bit stuck on this question. Could anyone share some ideas? Note: $K$ is the field I believe from the definition of the $ker(T)$ we can tell $n = 3$, but I am unsure as to how ...
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655 views

Eigenvalues of projector matrix

The spectrum of a projection is contained in $\{0, 1\}$, as $$(\lambda I-P)^{-1} = \frac{1}{\lambda} ( I - P) +\frac{1}{\lambda-1 }P$$Only $0$ and $1$ can be an eigenvalue of a projection, the ...
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Understanding the Frobenius Norm for Sparse Coding

I have a question regarding sparse coding, Non-negative sparse coding. Iterate until convergence: $ \mathbf{A_i} \leftarrow \arg \! \min_{A \geq 0} || \mathbf{X}_i - \mathbf{B}_i\mathbf{A}||_F^2 + ...
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38 views

Finding a pair of Orthogonal Vectors

Want: Pair of orthogonal vectors in $R^4$ that are also orthogonal to the vector (1,1,-2,3) My attempt at a solution: I got stuck...
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2answers
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how to combine angle rotations along different axes into one rotation along a single vector [duplicate]

So, lets say I have some rotation a about the x-axis(vector:$(1, 0 ,0)$) and some other rotation about y-axis(vector $(0, 1, 0)$) and a rotation about the z-axis(vector: $(0,0,1)$). How would I ...
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3answers
506 views

Does matrix addition give you a matrix or a number?

I am very confused by something our lecturer said today: We were given two matrices: $B=\begin{pmatrix}2 & 3\\ 2 &0 \\ 0&3\end{pmatrix}$ C=$\begin{pmatrix}6 ...
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1answer
63 views

Is this special matrix invertible?

The symmetric, tridiagonal $n-$by$-n$ matrix with the elements $a_{ii+1} = a_{i+1i} $ and off-diagonals' absolute values equal to the diagonal (except for row 1 and row n) is invertible. The elements ...
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1answer
27 views

What's the best way to think about the covariance matrix?

Let $X$ be a random vector with covariance matrix $\Sigma$. People often describe $\Sigma$ in terms of its components: $\Sigma_{ij}$ is the covariance of the $i$th and $j$th components of $X$. But ...
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Showing that $f_0 (x_1, \ldots, x_m) \mathrm tr A = \sum_{i=1}^n f_0(x_1, \ldots, Ax_i,\ldots, x_m)$

Question: Consider $f: (-\epsilon, \epsilon) \to \mathbb R^{m^2}$ a differentiable path of matrices $m \times m$ such that $f(0) = I_m$ and the function $g: I \to \mathbb R$ is defined by $$g(t) = ...
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Determine point of interesction of plane with axis given points of plane

Q: The points $(2,-1,-2)$, $(1,3,12)$ and $(4,2,3)$ lie on a unique plane. Where does the plane cross the z-axis. I understand that the point of intersection would occur at $(0,0,z)$ and I have ...
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2answers
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Invariant subspaces of the identity map

How many invariant subspaces does the identity map on $R^2$ have? My attempt: {0}, which coincides with the kernel. $R^2$, which coincides with the image and eigenspace. Is that it?
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$ det(A).\lim_{t \to 0} \frac{det(Id+tA^{-1}X)- det(Id)}{t} =tr (A^{-1}X)$

In some note it is written that $$ det(A).\lim_{t \to 0} \frac{det(Id+tA^{-1}X)- det(Id)}{t} =tr (A^{-1}X)$$ I could not understand how this is happen. Can someone explain it in detail please.
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Product of orthogonal projections

I need an example of two orthogonal projections such that their product is not a projection. I'm aware of this: Product of orthogonal projections need not be a projection Unfortunately, I've no idea ...
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Finding the decomposition of a function on a cubic spline basis of functions

In a computational project, I need to solve a partial differential equation. Standard procedure is to consider the weak formulation of the problem which maps it onto an algebraic problem. With cubic ...
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runninig pseudoinverse of a wide possibly rank-deficient matrix

There is a wide matrix $A=[a_1|a_2|..a_n]$ with m rows and n columns where $m<n$ where $a_i$ is the $i_{th}$ column vector of $A$. Let $A'$ be $A=[a_2|..a_n|a_{n+1}]$. Is there an efficient way of ...
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Finding individual vectors given a dot product or vector length

$$u \cdot v = 2\\ v \cdot w = -6\\ u \cdot w = -3\\ ||u|| = 1\\ ||v|| = 2\\ ||w|| = 7\\$$ (a) $<2v - w, 3u + 2w>$ If I'm given a vector length or dot product like the above how can I find the ...
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62 views

How to find bases for a subspace defined by an equation

How would you find the bases for $W_1 = \{(a_1,a_2,a_3,a_4,a_5) \in F^5: a_1 - a_3 - a_4 = 0 \}$ and $W_2 = \{ (a_1,a_2,a_3,a_4,a_5) \in F^5: a_2 = a_3 = a_4$ and $a_1+a_5 = 0 \}$ ? Would the basis ...
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Can we treat Covariance matrix as linear transformation.

This question is related to the link: What is the difference between matrix theory and linear algebra? My question is: when we see a matrix in any equation how to determine if that matrix is a ...
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prove that a system $AX=Y$ has soloutions iff the row rank of A is same as that of the augmented matrix?

Let A be a m by n matrix over the field F , I want to prove that a system $AX=Y$ has solutions iff the row rank of A is same as that of the augmented matrix of the system My try: I was thinking that ...
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28 views

Permutation as a product of generators of the permutation group

Let $G$ be a permutation group, generated by $g_1,\ldots,g_n$. And let $h$ be in $G$. Example: $G=\langle (12)(34),(123)\rangle$ and $h=(12)(34)(123)=(243)$ (reading the cycles from right to left, ...
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Does the inverse of a polynomial matrix have polynomial growth?

Let $M : \mathbb{R}^n \to \mathbb{R}^{n \times n}$ be a matrix-valued function whose entries $m_{ij}(x_1, \dots, x_n)$ are all multivariate polynomials with real coefficients. Suppose that ...
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1answer
21 views

Let W be the collection of all 2 by 2 symmetric matrices. Describe the orthogonal complement of W. (please)

A matrix is symmetric if $A^T$=A And the standard basis for symmetric matrices is [a,b], [b c] written as rows of a 2x2 matrix (sorry don't know how to make a matrix on this site). My question: How ...