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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Showing that a bilinear form is non degenerate

Given a finite dimensional vector space $V$ over $F$ and a fixed matrix $(\alpha_{ij})=A \in M_n(F)$ and the bilinear form on $V \times V$ by $B(u,v)=\sum_{i=1}^{n} \sum_{j=1}^{n} \alpha_{ij} \zeta_i \...
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42 views

What does it mean to find a basis that “diagonalizes” a transformation?

I'm having a hugely hard time wrapping my head around this statement. I am trying to figure it out on my own but I just don't get it. The terminology is weird to me and I can't really picture what it "...
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84 views

Linear Algebra requirement Spivak's Calculus on Manifolds

I am interested in the extent of knowledge of Linear Algebra required for Spivak's Calculus on Manifolds. More precisely, in the first problems in his book they reference norm preservation and inner-...
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172 views

Prove line-point distance formula in 2D

In the case where the line is in 2D and its equation has the general form $ax +by =c$, the distance $d(B, l)$ from $B = (x_0, y_0)$ is given by the formula: $$d(B, l) = \frac{|ax_0+by_0-c|}{\sqrt[]{...
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43 views

How to decide if solution exists for a linear equation?

I have $p$ ( $P_1,P_2...P_p$ ) positions and $n$ ( $N_1,N_2...N_n$ ) options to fill each position. Thus I have $n^p$ $p$ length strings. Each of these strings has a variable corresponding to them $\...
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27 views

Inverse of a linear transformation

What is the inverse of the following linear transformation? $T^{\theta}:R^2\rightarrow R^2$ a reflection in the line through the origin which forms an angle $\theta$ with the $x$-axis. I ...
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131 views

Span of a linear mapping

Let $ L : \mathbb{R}^n \to \mathbb{R}^m$ be a linear mapping such that $\text{rank}(L) = m$. If $\{v_1, \dots , v_k\}$ spans $\mathbb{R}^n $, then $\{L(v_1), \dots, L(v_k)\}$ spans $\mathbb{R}^m$. I ...
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34 views

Approximate sparse linear solve

I have a linear system $Ax=b$ that I need to solve many times with the same $A$ but slightly different $b$. $A$ is large, sparse, and SPD. Since this is for 3D rendering software, error requirements ...
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55 views

Taking limits of norms of a matrix raised to the nth power:

Given a matrix $$ A = \begin{bmatrix} 0 & 3 \\ -2 & 5 \\ \end{bmatrix} $$ and a vector $x = \begin{bmatrix}1&0\end{bmatrix}$, compute $\lim_{n\to\...
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42 views

Basis for the linear spacer $\ell^p$ [duplicate]

Is there any well known basis (Hamel basis) for the vector space $\ell^p$? And what about the cardinality of such basis? Is it countable?
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Linear transformation and bilinear forms

I am trying to do the next exercise (Exercise 11, section 10.2, Hoffman & Kunze). I defined a linear functional $L: V \to V^*$ as $(L_f\alpha)(\beta)=f(\alpha, \beta)$. It is easy to show that ...
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Matrix Representation $[T]_\beta^\alpha$ of $T:\mathbb{R}^3 \to P_2(\mathbb{R})$

We have the transformation $T:\mathbb{R}^3 \to P_2(\mathbb{R})$ defined by $T(a_1, a_2, a_3) = (a_1 + a_2 + a_3) + (a_1 - a_2 + a_3)x + a_1x^2$. We know the standard ordered basis of the domain $\...
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29 views

Finding a matrix representation of the linear transformation $T\colon P_2\to P_2$ ($T(f) = f''+2f'-f$)

Find a matrix representation of the linear transformation $T: P_2( \mathbb{R} ) \to P_2(\mathbb{R} )$, where $T$ is defined as $T(f(x)) = f''(x)+2f'(x) -f(x)$. I know the standard ordered basis of $...
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1answer
50 views

Method for ?not quite? weighted least squares fitting for more realistic results

I need a linear least squares type of fitting algorithm that understands how to weight the probability of a response coming from certain functions over another. To explain, given the standard linear ...
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24 views

Overdetermined system with discrete data.

The setup I have a set of experimental data (subscript 1) which calculates two variables $u_1(x,y,z)$ $v_1(x,y,z)$ I can calculate the three spatial gradients for my two variables ($u_1$ and $v_1$...
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49 views

prove image of quadratic form is $\mathbb{R}$

Let $V$ be a linear space over $\mathbb{R}$. Show if $q:V \to \mathbb{R}$ is quadratic form of rank $n=\dim V$ and there exist non zero vector $v \in V$ such that $q(v)=0$ then $q[V]=\mathbb{R}$.( [] ...
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1answer
188 views

Using a determinant to find the Cartesian equation for a plane from its parametric equations

This horribly unreadable webpage describes a method to find the Cartesian equation for a plane given its parametric equations. I'll try to type the method out here in a neater fashion: The ...
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1answer
33 views

Using the definition of unitary / orthogonal operators explicity for matrices:

If A is unitary, then $$AA^* = A^*A = I, and\ A^* = A^{-1}$$ I want to see this explicitly for a very simple unitary matrix, say, take the column vector A = (1,0,0) and we regard this as a 3x1 ...
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25 views

3 dimensional matrices and vectors

Let A be a $3 \times 3$ matrix such that $A\begin{pmatrix}3\\4\\5\end{pmatrix}=\begin{pmatrix} 2 \\ 7 \\ -13 \end{pmatrix}, A \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix} = \begin{pmatrix} -6 \\ 0 \\ 4 \...
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41 views

matrix norm equality

I've come across the following equality in a Linear Algebra book: For nonsingular $A \in \mathbb{C}^{n\times n} $, and $w \in \mathbb{C}^n $ $$\max_{w} \frac{\|w\|}{\|Aw\|}=\max_{w} \frac{\|A^{-1}w\...
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176 views

How to decompose matrix transformations

Let us assume $A$,$B$ and $C$ are known affine transformation matrices in homogeneous 2D space. If it should happen that $C=A^m B^n$ for some unknown $m,n$, is there a way to detect this short of ...
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Can a curve represented by a polynomial equation having n arbitrary constants, be uniquely determined by n-1 distinct points?

Let there be a polynomial $f(X)$, $X \in R^m$ with $n$ unknown arbitrary constants. Consider , the equation $ f(X) = 0$. Now, given $k$ distinct points in $R^m$ which satisfies the aforementioned ...
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Generalized criterion for positive definiteness of real symmetric matrices [duplicate]

I searched for the following question on stack exchange but couldnot find the general case answered or asked anywhere. This is an exercise from Artin's book, under the chapter named "Bilinear forms''. ...
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52 views

Construct and apply a rotation matrix by doing the following:

I'm having a hard time with starting this question: Create a $2 \times 2$ rotation matrix $A$ that is different from $I$.
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30 views

dependent eigenvectors for one λ

If we have a matrix and it has two eigenvalues $\lambda_1$ and $\lambda_2$ with $\text{mult}(\lambda_2)=2$, vector $v_1$ corresponds to $\lambda_1$ and vectors $v_2$ and $v_3$ correspond to $\lambda_2$...
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1answer
40 views

Is $B^TAB>0$ if $B>0$ and $A >0$?

If matrix $A$ is positive definite and matrix $B$ is positive definite, is $B^TAB$ positive definite?
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81 views

When does a complex matrix have a square root?

What is the neccessary and sufficient condition for a matri$A \in C^{n \times n}$ so that it has a "square root" , that is there is a matrix $B \in C^{n \times n}$ such that $A=B^2$. I am not quite ...
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78 views

Is there any relation between the Gershgorin circles of a matrix and its resolvent?

Let $A$ be a real symmetric matrix. Now fix a diagonal index say "i" and let $x > max-eigenvalue(A)$. Now is there any thing known about the Gershgorin circle of $[1/(x-A)]_{ii}$ in terms of the $A$...
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311 views

Projection of a vector onto the null space of a matrix

I have the following optimization problem: $$ \text{minimize}_x \Vert z - x \Vert^2 \\ \text{subject to } Ax = 0, $$ where $x,z\in \mathbb{C}^N$, and $A\in\mathbb{C}^{M \times N}$. $A$ is a wide ...
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110 views

Matrix with non-negative eigenvalues

Let $A \in \mathbb{R}^{n \times n}$ be a positive semi-definite $A \succcurlyeq 0$, and with positive diagonal elements ($A_{i,i} > 0$ for all $i$). Let $A$ have at least one eigenvalue equal to $0$...
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167 views

Give a formula for the volume of the solid under a surface $z=xy$ and a triangle?

Given is the solid with unit density lying under the surface $z = xy$ and above the triangle in the $xy$-plane with vertices $(0, 1, 0)$, $(1, 1, 0)$ and $(0, 2, 0)$. Give a formula for the ...
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Expectation of Projection Matrix

I'm interested in the following question: Let $\mathbf{A}\in\mathbb{R}^{N\times M}$, $M\leq N$ be an random matrix with i.i.d. entries that take strictly positive values. A continuous probability ...
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167 views

Is this dual pairing the same as the inner product?

If $(V, \langle \cdot$ , $ \cdot\rangle)$ is an inner product space with dual $V^*$ then there is a natural dual pairing $\langle \cdot$ , $ \cdot \rangle ^*: V^* \times V \rightarrow \mathbb K$ given ...
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Checking if a matrix is positive matrix after some coordinate transformations

I have the next positive definite matrix $Q\in\mathbb{R}^{3\times 3}$. For a full rank $U$ (e.g. defining a linear coordinate transformation) I can decompose $Q$ as $$ Q = U^{-1}TU, $$ where obviously ...
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47 views

Help on understanding this process of getting orthonormal basis

This method was taught by someone and I am trying to understand the process. Assume that we have $(1,2,3)$ as first vector. Since we need another vector that will make the dot product $= 0$ with the ...
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1answer
40 views

Set of a matrix

I am working on a homework problem which asks me about the Set of a singular $n\times n$ matrix. specifically whether it is a vector space. I looked in the glossary of the book and searched online and ...
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337 views

Kernel of a bilinear map and tensor product specificially

I am cementing my understanding of tensors and a book I am reading handwaves and simply says "of course we may denote $0\otimes 0$ as the $0$ of $U\otimes V$" I have proved that the kernel is larger ...
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45 views

vector as production of matrices, trouble with exp()

A vector $(a_{11}x, a_{22}y, a_{33}z)$ can be expended as: $$\begin{align} \begin{pmatrix} a_{11}x \\ a_{22}y\\ a_{33}z \end{pmatrix} &= \begin{pmatrix} a_{11} & a_{12} & a_{13}\\ ...
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1answer
53 views

Is the conjecture $Per A $ is the largest eigenvalue of $\tilde{A}$ being solved?

Let A be a positive semidefinite matrix of order $n$. Is the conjecture $Per A $ is the largest eigenvalue of $\tilde{A}$ is being solved? Where $\tilde{A}$ is the matrix of order $n!\times n!$...
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34 views

Finding a line parallel to another one, but going through a point not on the line?

Is it possible to have two parallel lines of the form $$x=x_0 + tv$$ where one line in vector equation form that has a point $x_0$ that does not lie in the line of the other line that it is parallel ...
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36 views

Eliminate matrix?

Does $A \cdot$ $A^T$ cancel each other? I am partly trying to understand how linear least square works.. I see that that the projection is the closest solution given as $Ax^*$ but do not see why $A^T$...
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42 views

Sum of Matrix Norms and boundedness

Given a matrix $N$, show that there is a constant $C$ such that $$||I + tN + \cdots + t^n N^n|| \le Ct^n$$ for all sufficiently large $t$. I am not sure how to show this. I am guessing I am supposed ...
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155 views

Characteristic polynomial of a tridiagonal matrix

Consider the polynomial recurrence $$p_{k+1} (x) = (x - \alpha_{k+1})p_k(x) - \beta^2_{k+1}p_{k-1}(x), \quad (k=0,1,\ldots)$$ where $p_0 = 0$, $p_{-1}=0$, and $\alpha_k$ and $\beta_k$ are scalars. ...
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does $\|A\| > \|B\|$, $A$ and $B$ matrices, imply that $\|Ax\| >\|Bx\|$ for all $x$ in some vector space?

I'm wondering if I can prove the first inequality in a question that I am working on, does that make the second inequality automatic? Thanks,
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43 views

Given $A$ and $B$ are matrices (or transformations), what is the difference between “$A$ is similar to $B$” and “$A$ is isomorphic to $B$”?

I am trying to find a property (either similarity or isomorphism or other) that describe when two matrices are the same. More particularly, I am trying to describe adjacency matrices that are the same ...
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282 views

Why does a symmetric matrix have a complete set of eigenvectors and eigenvalues?

I am attempting to learn more about the adjacency matrix(graph theory) but given that I have forgotten a lot of linear algebra, I can't seem to know why this is true. Can someone give me a proof?
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Non-computational proof that $\det(A)$ is a unit in $R$ implies $A$ is a unit, for $A \in M_n(R)$.

Quoting from Waterhouse's Introduction to Affine Group Schemes: "...suppose we have a representable functor $G$ [from $k$-algebras to groups], and a map $\Delta: A \to A \otimes A$ giving a ...
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2answers
202 views

Vectors parallel to plane, perpendicular to another vector

From Anton, I have this simple looking LA question: Find all unit vectors parallel to the yz plane that are perpendicular to the vector 3,1,-2 Since this vector is sloped in all 3 dimensions, and ...
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1answer
36 views

Representing real numbers from matrices of non-negative-reals.

Consider $I = \left(\begin{array}{cc} 1&0\\0&1\end{array}\right)$ and $N = \left(\begin{array}{cc} 1&1\\1&1\end{array}\right) - I = \left(\begin{array}{cc} 0&1\\1&0\end{array}\...
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2answers
65 views

The Standarization of Matrix by Vector Multiplication

I apologize for the trivialness of my question but it has been bugging me as to why the standard for multiplying a matrix by a vector that will give a column matrix mean that the vector has to be a ...