Articles of linear algebra

Isomorphism with dual space

Let $V$ be a vector space over $F$ with basis $\{v_1,\cdots, v_n\}$, and for every $i$, let $f_{v_i}\colon V\rightarrow F$ be a linear map satisfying $f_{v_i}(v_j)=\delta_{ij}$. Then $\{f_{v_1}, \cdots, f_{v_n}\}$ forms a basis of the dual space $V^*$. The map $T\colon v_i\mapsto f_{v_i}$ uniquely extends to a linear injective map from $V$ to $V^*$, and […]

Condition for mapping linearly independent vectors to linearly independent vectors

Let $f \colon U \to V$ be a linear map and $\{u_1, u_2, \ldots, u_n\}$ be the set of linearly independent vectors in $U$. Then the set $\{f(u_1), f(u_2),\ldots,f(u_n)\}$ is linearly independent iff a. $f$ is one-one & onto b. $f$ is one-one c. $f$ is onto d. $U = V$ I came to conclusion […]

difference between parallel and orthogonal projection

i would like to understand what is a difference between parallel and orthogonal projection?let us consider following picture we have two non othogonal basis and vector A with coordinates($7$,$2$),i would like to find parallel projection of this vector to these basis,i am studying Covariant and Contrivant components,so i would like to understand how to find […]

Easiest way to solve system of linear equations involving singular matrix

I am trying to balance an unbalanced chemical equation by using setting up a system of linear equations to solve for the stoichiometric coefficients in the chemical equation. After setting up a matrix, to try and solve the system, i cant because one of the matrices is a singular array. I have taken a look […]

Cross product as result of projections

The cross product between two vectors in $\Bbb{R}^3$ (call them a and b) is denoted a $\times$ b and the result is a vector in $\Bbb{R}^3$ orthogonal to the first two. There are a variety of ways of computing this resultant vector. One way in particular is known from the symbolic determinant involving i j […]

Let $W$ be a subspace of a vector space $V$ . Show that the following are equivalent.

Show that $\textbf{v} + W \subseteq W \Rightarrow W \subseteq \textbf{v} + W.$ Here is my proof, is it correct? Is there any easier way? Note that for any element $\mathbf{v + w}$ in $v+W$, $\mathbf{v+w}$ is in $W$ as well. Hence $\mathbf{v} + \mathbf{w} = \mathbf{w}_{1}$ for some $\mathbf{w_1} \in W$ Which implies $\mathbf{v} […]

unique factorization of matrices

If I have a set of matrices, call this set U, how can I make this a UFD (unique factorization domain)? In other words, given any matrix $X \in U$, I would be able to factorize X as $X_1 X_2 … X_n$ where $X_i \in U$ and this factorization is unique? We may assume the […]

Priority vector and eigenvectors – AHP method

I’m reading about the AHP method (Analytic Hierarchy Process). On page 2 of this document, it says: Given the priorities of the alternatives and given the matrix of preferences for each alternative over every other alternative, what meaning do we attach to the vector obtained by weighting the preferences by the corresponding priorities of the […]

$SL(3,\mathbb{C})$ 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 $\mathbb{C}^3$. What I’m having trouble showing is that mapping $g$ to $h(g)$ is a group homomorphism. Namely, I know that $h(ab)p(z)=p(b^{-1}a^{-1}z)$, but I can’t […]

Show the negative-definiteness of a squared Riemannian metric

Let $\Bbb{S}_{++}^n$ denote the space of symmetric positive definite (SPD) $n\times n$ real matrices. The geodesic distance between $A,B\in\Bbb{S}_{++}^n$ is given by the following Riemannian metric $$ d(A,B):= \Bigg(\operatorname{tr}\bigg(\ln^2\big(\sqrt{A^{-1}}B\sqrt{A^{-1}}\big)\bigg)\Bigg)^{\frac{1}{2}}. $$ EDIT It could also be defined as follows $$ d(A,B):= \lVert\log(A^{-1}B)\rVert_{F} $$ EDIT II I could show the negative-definiteness of $d^2$ if I could write […]