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 ...

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Basis for a linear map

Let $V$ be a vector space over $\mathbb C$ with $\dim V=n$ and $F\colon V\to V$ be a linear map. (a) Show that there always exists a basis $\{v_1,\ldots,v_n\}$ such that $F(v_j)$ is in space $W_j$ ...
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388 views

Vectors / Linearly independent question

an easy question that I completely understand, just not sure how to algebraically prove. $u,v,w$ are vectors in $R^3$ given $u\times v +v\times w + w\times u=0$ I need to prove that {u,v,w} are ...
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122 views

linear maps on complex inner product spaces

Problem: $W$ is a complex inner product space. $A$ is a linear map defined on $W$ that satisfies: $\left \langle x,y \right \rangle=0\Rightarrow \left \langle Ax,Ay \right \rangle=0$ for any $x,y\in ...
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297 views

Basis-free formulation of Jordan normal form theorem

Is there a basis-free formulation of Jordan normal form theorem? From some search I did in Google, the answer is apparently yes. But I didn't find any article that I could understand. (I've only ...
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79 views

matrix multiplication, confusion

Am I correct to say that this matrix $C$ cannot be found $$C\times\left(\begin{array}{cc} 9 & 1\\ 4 & 6\\ 3 & 4\end{array}\right) = \left(\begin{array}{cc}9&1\\4 & 6\\ ...
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115 views

Approximating the Hessian

Theorem. Let $B_k$ be a symmetric matrix. Let $B_{k+1} = B_k+C$ where $C \neq 0$ is a matrix of rank one. Assume that $B_{k+1}$ is symmetric, $B_{k+1}s_{k} = y_k$ and $(y_{k}-B_{k}s_{k})^{T}s_{k} ...
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119 views

How to solve a linear algebra homework problem?

Let $K$ be a field, suppose that $D\colon M_{n\times n}(K) \to K$ is a function such that $D(AB)=D(A)\cdot D(B)$ and $D(I) \neq D(0)$, where $0$ is the zero matrix. Show that if ...
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60 views

A set of fixed points

How can we go about finding a Moebius map that fixes the set $\{z_1=x+iy,\,\,\, z_2={1\over iy-x}\}$ for some $x,y\in \mathbb R$ that does not correspond to rotation about any arbitrary axis of the ...
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74 views

Reordering an unknown linear algebra theorem and proof into the correct order

Good day everyone, I have got a little puzzle that needs solving. A friend has given me a theorem and its proof which has been jumbled-up with two pieces introduced into it that do not belong in the ...
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347 views

Riemann sphere and Maps

Could somebody please clarify the following for me? I am not too clear about the relationship between the Riemann sphere and Möbius maps. I know that we can through projection make some Möbius maps ...
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I'm interested in different meanings of “normal”~ [duplicate]

Possible Duplicate: What is it to be normal? I've learned in algebra class that "normal" means a linear operator is commutative with its adjoint; also we say that $H$ is a normal ...
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Calculate the vector normal to the plane by given points

How can one calculate the vector normal to the plane that is determined by given points? For example , given three points $P_1(5,0,0)$, $P_2(0,0,5)$ and $P_3(10,0,5)$, calculate the vector normal to ...
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243 views

“Algebraic multiplicity” for eigenvalues of a Sturm-Liouville-like problem?

Following Coddington-Levinson's book Theory of ordinary differential equations, chapter 7: "Self-adjoint problems on finite intervals", let us consider the eigenvalue problem $$\pi(l):\begin{cases} ...
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220 views

set of all trace zero matrices with bounded entries in $M_2(\mathbb{R})$

$X=\text{set of all trace zero matrices with bounded entries in } M_2(\mathbb{R})$ and $Y=\{detA: A\in X\}\subseteq\mathbb{R}$ Does there exist $\alpha<0$ and $\beta>0$ such that ...
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101 views

When do the invariant factors of a direct sum of diagonal matrices correspond to those of its summands?

I am trying to prove something about matroids, which I have reduced to the following question: Suppose I have a matrix $M$ which is a direct sum of submatrices $M_1,M_2,\ldots,M_k$. When do the ...
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307 views

Compactness of the set of all unitary matrices in $M_2(\mathbb{C})$

Is the set of all unitary matrices in $M_2(\mathbb{C})$ is compact? I can show that as determinant map is continuous so unitary matrices are closed but how to show they are bounded? Please help.
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483 views

Computing the rank and signature of a quadratic form - quick way?

Is there a 'quick way' of computing the rank and signature of the quadratic form $$q(x,y,z) = xy - xz$$ as I can only think of doing the huge computation where you find a basis such that the matrix of ...
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106 views

Comparing the spectrum of the (non)centered matrices

Suppose a symmetric matrix $A\in\mathbb{R}^{n\times n}$ is given. Let $J=I-\frac{1}{n}\cdot 1_n1_n^T\in\mathbb{R}^{n\times n}$ be the centering matrix, with $I$ being the identity matrix, and $1_n=[1 ...
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35 views

Specific range of numbers is given, trying to get another number within same range

I'm trying to calculate the width of an HTML element based on the window size. Here's what I have. These width values (first value) accurately match with the width the HTML element must be (second ...
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492 views

Listing all the albelian groups of order 900

A past exam question says: Using the fundamental theorem (of finitely generated albelian groups), list all the abelian groups of order 900 where no two groups in your list should be isomorphic but any ...
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122 views

Derivative/Chain Rule (for MANLYfolds) Computation

Embarrasingly, I can't compute the following derivative. $dh(X)=\left.\frac{d}{dt}h(e^{XT}]\right|_{t=0}$, where $X$ resides in the lie algebra of $\rm SL(3,\Bbb C)$ [ie $\mathfrak{sl}(3,\Bbb C)$] ...
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Matrix Exponential with time parameter

Could someone please expand on Method 9. Lagrange interpolation (page 17) at http://www.cs.cornell.edu/cv/researchpdf/19ways+.pdf because the summation runs from 0 to (n-1) but the eigenvalues ...
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Projecting onto subspace

Question Find the orthogonal projection of $$x = \begin{bmatrix}7 \\ 0 \\ -4 \\ -4 \end{bmatrix}$$ onto the subspace of $\mathbb R^4$ spanned by $$v_1 = \begin{bmatrix}-4 \\ 2 \\ 2 \\ -4 ...
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Lie Algebra Homomorphism Question

So this is a bit of a follow-up to my recent question. I don't mean to inundate the feed with my quandaries, but as I move through the theory I keep hitting stumbling blocks (which y'all so kindly ...
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66 views

Area of a geometric configuration

How to find the area of the triangle in the plane R2 bounded by the lines y=x, y=-3x+8 and 3y+5x=0. How can I solve this? I'm thinking i can take y=x as the origin and just use y=-3x+8 and 3y+5x=0 ...
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419 views

Why is Cholesky factorization numerically stable

It's often stated (eg: in Numerical Recipes in C) that Cholesky factorization is numerically stable even without column pivoting, unlike LU decomposition, which usually need pivoting schemes. But ...
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1answer
155 views

Orthogonal basis for $P_{2}(\mathbb{R})$

Consider $P_{2}(\mathbb{R})$ together with inner product: $$\langle p(x), q(x)\rangle = \int_{0}^{1} p(x)q(x) \, dx$$ I am trying to come up with an orthogonal basis with respect to this inner ...
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749 views

Definition of isomorphism of vector spaces

Suppose that $V_1$ is a vector space over the field $K_1$ and $V_2$ is a vector space over the field $K_2$. What is the definition of an isomorphism between these two vector spaces? My best guess ...
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384 views

Convex hull of sets defined by (in)equalities

If you define the convex hull of a set $X$ as the set of all convex combinations of elements of $X$, it becomes difficult to decide if a given element $w$ belongs or not to $conv(X)$ (You have to ...
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125 views

Is an orthonormal set of vectors implied to be orthogonal?

Is an orthonormal set of vectors implied to be orthogonal? Why do they call the matrix=QR orthogonal and not orthonormal ?
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446 views

Finding possible minimal polynomials from a characteristic equation.

A question says, write down the possible minimal polynomials which have characteristic polynomial $(1-x)(1-x^3)$, and for each possibility find a specific example of a matrix having this minimal ...
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3answers
461 views

Can a matrix in $\mathbb{R}$ have a minimal polynomial that has coefficients in $\mathbb{C}$?

Like the title says, can a matrix in $\mathbb{R}$ have a minimal polynomial that has coefficients in $\mathbb{C}$? I have a feeling the answer is yes because the characteristic polynomial can have ...
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1answer
92 views

$SL(3,\mathbb{C})$ acting on Complex Polynomials of $3$ variables of degree $2$

So I'm given the following definition: $h(g)p(z)=p(g^{-1}z)$ where g is an element of $SL(3,\mathbb{C})$, $p$ is in the vector space of homogenous complex polynomials of $3$ variables and $z$ is in ...
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748 views

why does the blockwise inversion formula work?

I used this http://en.wikipedia.org/wiki/Matrix_inverse#Blockwise_inversion formula to get the inverse of a partitioned matrix, and it works great. What I don't understand is why exactly it works. If ...
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389 views

program for eigenvalue calculation

I have a n x n matrix. I would like to (a) take successively higher powers of the matrix and then multiply by projection vectors until the resulting vectors differ by only a scalar factor. (b) ...
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What is a basis for the vector space of continuous functions?

A natural vector space is the set of continuous functions on $\mathbb{R}$. Is there a nice basis for this vector space? Or is this one of those situations where we're guaranteed a basis by invoking ...
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957 views

Finding Bases for Kernel and Image of Linear Transformation

I'm trying to solve a linear transformation problem. Let $ \psi: \mathbb R_3 [x] \to \mathbb R_4 [x] $ be defined by $ \psi : p(x) \mapsto x^4 p(1/x)+p(x)$ Q) Show that $\psi$ is a linear ...
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548 views

Symmetric bilinear form positive definite?

$V$ denotes real vector space $V$ consisting of all polynomials with real coefficients/any degree so $V$ is infinite dimensional. A symmetric bilinear form is defined on V by $$(f,g) = \int_0^\infty ...
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1answer
65 views

Proof $||A||_{p} < 1 \Rightarrow \lim\limits_{k \rightarrow \infty}{A^k} = 0$ for any $A \in \mathbb R^{n \times n}$

Consider any matrix $A \in \mathbb R^{n \times n}$ with the p-norm $||A||_{p} < 1$. I would like to show that $\lim\limits_{k \rightarrow \infty}{A^k} = 0$. Consider the reverted scenario. Let ...
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1answer
100 views

For what matrices $A$ and $B$ is $\operatorname{tr}(AB) = \operatorname{tr}(A)\operatorname{tr}(B)$?

I was wondering for what matrices (over $\mathbb{C}$) $A$ and $B$ is the equation $\operatorname{tr}(AB) = \operatorname{tr}(A)\operatorname{tr}(B)$ satisfied?
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2answers
85 views

Have spd $(A^TA)$ and $(B^TB)$, need $A^TB$.

Given two symmetric positive definite matrices $(A^TA)$ and $(B^TB)$ I need to compute $A^TB$. $A$ and $B$ are not given directly. $(A^TA)$ and $(B^TB)$ have the same dimensions. $A$ and $B$ are ...
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180 views

The minimal polynomial written as a product

For $A \in \mathbb{C}^{n,n}$ and $\{ \lambda_1, \dots , \lambda_r\}$ are the eigenvalues of $A$. My lecture notes say that the minimal polynomial of $A$ is $$\prod_{i=1}^r(x-\lambda_i)^{a_i}$$ where ...
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Is the action transitive?

Let $G=GL(n, \mathbb{C})$ and $M=\{Q \in GL(n, \mathbb{C}) | Q^t = Q \}$. Suppose $G$ acts on $M$ through the following: $\forall g \in G, \forall m \in M$, $g\cdot m = gmg^t$. Question: Is the action ...
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87 views

Is $APA^t$ diagonalizable or not?

Let $G = GL(n, \mathbb{C})$ and $M = \{Q \in GL(n, \mathbb{C}) | Q^t = Q\}$. Let $P\in M$ be non-diagonalizable. Question: For any $A \in G$, can we say that $APA^t$ is also non-diagonalizable? Or ...
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1k views

Dot product of two vectors without a common origin

Given two unit vectors $v_1, v_2\in R^n$, their dot product is defined as $v_1^Tv_2=\|v_1\|\cdot\|v_2\|\cdot\cos(\alpha)=\cos(\alpha)$. Now, suppose the vectors are in a relation $v_2=v_1+a\cdot1_n$, ...
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99 views

Questions regarding the column and row spaces of Echelon form

I have several conceptual questions that have been confusing me for a while in linear algebra. Let A be a 3 by 5 matrix with full rank rows. A is now simplified to Echelon form U and further ...
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70 views

Understanding a statement about the existence of functionals

Thm. Let $L$ be a solvable subalgebra of $gl(V)$, $V$ a finite dimensional nonzero vector space. Then $V$ contains a common eigenvector for all the endomorphisms in $L$. The proof of this theorem is ...
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163 views

is $U^{2}=P$? where $U$ unitary and $P$ orthogonal projection

Problem: I am trying to solve the following problem, but I couldn't. The problem is: Let $U$ be unitary matrix. Let $P$ and $UP$ be orthogonal projections. Is it true that $U^{2}=P$? If yes, please ...
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664 views

Countability of the zero set of a real polynomial

This is the question from my calculus homework: Is it possible for a polynomial $f\colon\, \mathbb{R}^{n}\to \mathbb{R}$ to have a countable zero-set $f^{-1}(\{0\})$? (By countable I mean countably ...
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3answers
194 views

Multiplicity of eigenvalues

Suppose $A$ is an $n\times n$ complex matrix. How to show the following two properties If $\lambda$ is an eigenvalue of $A\bar{A}$, so is $\bar{\lambda}$. Here $\bar{A}$ means the entrywise ...