# Tagged Questions

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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### A question about linear transformations mapping straight lines to straight lines

Obviously, a linear transformation over a space maps all straight lines to other straight lines. My question is: is the converse true? That is, if we're looking at a space after some transformation ...
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### How do I find 2x2 orthonormal diagonalizing matrices using only trigonometry?

I have a matrix $A=\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ (where all values are known), and I eventually want to diagonalize it into: $$A=UDV^T$$ for orthonormal U and V. If I ...
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### relation between isomorphism induced by Kahler form and any other (non-degenerate) two-form

Let $V$ be an euclidean vector space, and $\omega \in \wedge^2 V^*$ be non degenerate, i.e. the induced homomorphism $\tilde\omega: V \to V^*$ is bijective. What is the relation between the two ...
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In the numerical optimization text I am reading, the Steepest Descent Direction was derived by considering $$\min_{||p||_2\leq 1} p^T\nabla f(x_k)$$ This resulted in $$p_k=-\frac{\nabla f(x_k)}... 0answers 55 views ### Prove that a,b,c are the sides of a triangle a,b,c\in\mathbb R_{>0} are such that$$\begin{cases}a^2x+b^2y+c^2z=1\\xy+yz+zx=1\end{cases}$$has a unique solution (x,y,z)\in\mathbb R^{3}. Prove that a,b,c are the sides of a triangle. ... 1answer 99 views ### Computing intersection of vector spaces spanned by two lists Assume that I'm given two lists of vectors l_1 and l_2, where all the vectors have equal dimension. I want to compute a basis for the intersection of their spans. What is the easiest setup for ... 1answer 70 views ### Multiplicity as roots of the minimal polynomial Let V\neq\{0\} be a finite-dimensional vector space over a field F and let \alpha \in \text{End}(V). Suppose that \lambda is an eigenvalue of \alpha with multiplicity r as a root of the ... 0answers 287 views ### Gram-Schmidt orthogonalization basis for Continuous functions Use Gram-Schmidt orthogonalization process to find an orthogonal basis of the subspace of \mathbb{R}^4 spanned by v_1=(1,1,0,0),v_2=(1,1,1,0) \,and, v_3=(1,1,1,1) with respect to the (standard) ... 1answer 96 views ### Hamel Basis Exercise Proof Clarification. While looking up something else on stack exchange, I ran across this question An exercise about a Hamel basis and it intrigued me. The answer was provided by Jonathan Golan (http://math.... 0answers 23 views ### Linear algebra (Coordinates) Question: Find the coordinates of x=(1,0,0) in relation to base$$B=\{(1,1,1),(-1,1,0),(1,0,-1)\}.$$I tried: a,b,c\in R such that$$a(1,1,1)+b(-1,1,0)+c(1,0,-1)=(1,0,0)=x$$but I'm not sure ... 0answers 22 views ### Probablistic bound for \|RR^TM\| for uniformly random orthonormal matrix R I am stuck on a finding a probablistic bound on a nonstandard random matrix. I looked around on the internet and couldn't find any results. This could be because I don't know the key words or because ... 1answer 78 views ### Algorithm to determine if integer matrix is similar to symmetric integer matrix with nonnegative entries Let A\in M_n(\mathbb{C}) be a matrix with integer entries (treated as a matrix over the complex numbers). Is there an efficient way to check if A is similar to a symmetric matrix with nonnegative ... 1answer 123 views ### Partition of a Matrix In Linear Algebra, we have been taught that the partition of a matrix A consists of matrices,or blocks. In other words, its elements are matrices. This same, partitioned matrix, however is said to ... 0answers 266 views ### Vector space basis change: is this “index-free” notation correct? There are already quite a number of questions on basis change in a vector space. Nevertheless, to fully grasp the underlying idea I made up the following notation and I have some doubts on it (note: ... 0answers 26 views ### Find solution to matrix sandwich product [duplicate] For any two n \times n real symmetric and positive definite matrices B and C, is it always possible to find a third real symmetric and positive definite matrix A such that ABA=C? If not, ... 0answers 60 views ### Solving for matrices [T] and [T]', and the transition matrix Q given a basis. Problem Part 1: I've already proved that T is a linear transformation, but I need to verify that I solved the other parts of the problem correctly. Here's my attempt to find [T] and [T]' below: ... 0answers 160 views ### Constrained Quadratic Optimization(Reproducing Kernel) I am attempting to use a constrained quadratic optimization to find the coefficients of a reproducing kernel. The problem is as follows: y(t)=\sum_{i=0}^J\alpha_iK(t, t_i) Q(\alpha)= \alpha^tK\... 0answers 299 views ### Consistent Augmented Matrix Well, the linear system which at least has one solution is called "consistent" linear system. Find an equation involving g, h, and k that makes this augmented matrix correspond to a consistent system: ... 1answer 39 views ### Vector space property proof verification My problem is from Linear Algebra by Friedberg, Insel, Spence: In any vector space V, show that (a+b)(x+y)=ax+ay+bx+by for any x,y in V, and any a,b in F. My approach is:$$\begin{...
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There was a definition on my notebook. But sadly I cant read (...) part. What do we call $w_1,w_2,w_3...w_k$? Let V be a vector space on field F and $w_1,w_2, w_3..$ are subspaces of V. for any ...
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### Existence of a subspace with a certain property

I am having trouble solving this problem.I have started solving the problem , so far my guesses for the subspace U were the intersection of V and complement of KerT , but i was soon able to come up ...
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### Role of metric in the matrix representation of Hermitian adjoint

I'm working through Jeevanjee's "An Introduction to Tensors and Group Theory for Physicists", and while trying to prove that the matrix representation $M(A^\dagger)$ of a Hermitian adjoint $A^\dagger$ ...
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### Definite Integral involving matrices

We have a definite integral of the form given below $f(t) = \int_0^1 e^{\alpha X(t)} \frac{dX(t)}{dt} e^{(1-\alpha) X(t)}\,d\alpha \tag 1$ Given Data in the question $X(t)$ is a skew-...
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### Any idea how to linearize this equation? $X^2-Y^2=aZ+bZ^2$

The intention is to linearize this equation $X^2-Y^2=aZ+bZ^2$ into something which looks like $Z=mX+nY+c$ so that a graph of $Z$ against $X$ or $Y$ can be plotted. X,Y,Z are variables while a,b,c are ...
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### Equality in the sequence of increasing ranges.

Suppose $T \in L(V)$. Let $n = \dim V$. Prove that $\text{rangeT}^n = \text{rangeT}^{n+1} = \text{rangeT}^{n+2} = \dots$ I need help finishing this proof. This is what I have so far: First prove ...
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### Show that $(Au,Bv)=(u,A^tBv)$

Let $A, B$ be matrices of order $n$, and $\vec{u}, \vec{v}$ vectors from euclidean space $\mathbb{R}^n$, then $(Au,Bv) = (u,A^tBv)$ pd. $(\cdot ,\cdot)$ is my notation for inner product, ...
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### Convergence of square root operators

Let $Q_n$ and $Q$ be compact positive and symmetric operators. Let $A_n = {Q_n}^{\frac12}$ and $A=Q^{\frac12}$. Given $Q_n$ converges to $Q$ w.r.t. operator norm. Does $A_n$ converges to $A$? Thanks.
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### Consider a symmetric matrix $X$ with eigendecomposition $X=UVU^T$, how to call $\sum_{v_{k,k}>0}v_{k,k}u_ku_k^T$?

Consider a symmetric matrix $X$ with eigendecomposition $X=UVU^T$ How do people call $\sum_{v_{k,k}>0}v_{k,k}u_ku_k^T$? Sum of positive components of $X$? The positive semi definite part of $X$?
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### Find a linear operator given the kernel

"Find a linear operator $T:\mathbb{R}^3\to\mathbb{R}^3$ so that the kernel is generated by $(1,2,-1)$ and $(1,-1,0)$." It's been a while since I've worked with linear algebra, but from memory I know ...
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### Techniques to find matrix inverses of general classes of matrices?

Suppose you're given some general description of an $n\times n$ matrix, and asked to find its inverse. By "general description" I mean that the matrix can be described in one or more sentences, and ...
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### Existence of solution for matrix equation $(I - \alpha A) \bar{x}=\bar{b}$

This is my first question in here and I would be really thankful if someone could help me with understanding the matter. I am solving a matrix equation $(I-\alpha A) \bar{x} = \bar{b}$ for a positive ...
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### Recurrence - using power series

Could you help me in solving this recursion( a closed form ) using power series $\mu(n)=\mu(n−1)k_0+(n−1)\mu(n−2) k_1 \tag 1$, where $k_0,k_1$ are constants $\mu(0)=3,\mu(1)=5$ HINT: We can think ...
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### Least number not being the determinant of a set of matrices

Let n > 1 be a natural number and u < v integers. How can I determine the least natural number not being the determinant of some n x n - matrix with integers in the range u..v without calculating ...
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### Find a reduced echelon basis from a reduced echelon matrix.

The reduced row matrix was this ---> $\begin{pmatrix}1&2&0&1&0\\0&0&1&3&0\\0&0&0&0&1\\0&0&0&0&0&\end{pmatrix} = 0$ So i computed ...
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### Prove vertices of a simplex are affinely independent

I'm given that the definition of a simplex $T$ is $x \in\mathbb R^n$ such that $x$ satisfies $n+1$ linear inequalities: $(u_k, x) \lt c_k$ for $k = 1,\ldots,n+1$ (i.e. $T$ is the intersection of $n+1$ ...
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### Strange phenomena in determinants of matrix of determinants.

In my research, my computations are giving rise to the following strange phenomena: Let D=\begin{bmatrix}x_1^p & x_2^p & x_{3}^p\\ x_{1}^q & x_{2}^q & x_{3}^q\\ x_{1}^r & x_{2}^...
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### Analytic Geometry: One sheeted hyperboloid

Good afternoon! I have a question about analytic geometry. I don't actually know if the answer is quite simple, and I missed something while revising, or if it is actually more complicated than I ...
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### Finding gradient of an objective as a PDE

I am trying to find the gradient of the following optimization problem and then add to objective, but I got some trouble in computing. Could you please help me? Assume that we have an optimization ...
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### Line parallel to a plane and have 45 degrees between another

I need to find a direction vector for a line parallel to a plane $x+y+z = 0$ and that have $45$ degrees with the plane $x-y = 0$ So, i've assumed the vector $\vec V_r = (a,b,c)$ and since it is ...
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### Canonical embedding into dual space?

How would one go about proving that there is no embedding of a vector space into it's dual that is independent of a choice of basis? Thanks
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### Maximum determinant of a $m\times m$ - matrix with entries $1..n$

I want to find the maximal possible determinant of a $m\times m$ - matrix A with entries $1..n$. Conjecture 1 : The maximum possible determinant can be achieved by a matrix only ...
### If $\|Tv\|=\|T^*v\|$ for all $v\in V$, then $T$ is a normal operator
I have solved a question but I am not sure the last step of the question. If someone can verify it that would be great. Let $V$ be a finite dimensional vector space with complex inner product. Let ...
Can anyone help me with the following problem? Suppose $u=(u_1,u_2,...u_n)^T$, $e=(1,1,...1)^T$, and we have $u\geq e$. Now for any symmetric matrix $A\in S^n$ with $diag(A)=0$, can we claim the ...