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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Proof: linearly dependence for particular set in $\mathbb{R}^4$

The problem that I can't seem to solve is: Let $S$ be the set of all vectors in $\mathbb{R}^4$ with exactly $2$ entries equal to $1$ and all the rest of its entries equal to $0$. Is $S$ linearly ...
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1answer
705 views

Converting sum of infinity norm and L1 norm to linear programming

So I'm trying to convert this minimization problem, min $\parallel Ax-y \parallel_{\infty}$ + $\parallel x \parallel_{1}$ where $A$ is $m$ by $n$, $y$ is $m$ by $1$ and $x$ is $n$ by $1$. into a ...
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1answer
72 views

question about norms and convex set

Suppose $\overline{B}(0;1) = \{ x \in X : ||x|| \leq 1 \}$ is the closed unit ball on a vector space $X$. MY question is: is the following true? If $\overline{B}(0,1) $ is not convex, then $|| \cdot ...
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1answer
61 views

Linear Algebra Transformation Matrix

If the transformation is from $\mathbb{R}^3\to\mathbb{R}$ is $$T\{a,b,c\} = \int_0^\pi 2ae^t+2b\sin(t)+3c\cos(t)\, dt$$ How do I find the standard matrix? I'm not sure if I should solve this problem ...
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A question on inequalities

What is the solution set of the inequality $$ \frac{2x - 1 }{x+1}\lt0$$ One answer that is quite simple to get is $$x\lt1/2 $$ What can be the other value for the solution set...??
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1answer
83 views

Linear Algebra Span question

Let $a, b, c$ be vectors in $\mathbb{R}^3$. From what I understand, if $c\in \mathrm{Span}\{a,b\}$, then $b\in \mathrm{Span}\{a,c\}$. Since they all fall on the same plane, I can't seem to find a ...
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1answer
24 views

Factorizing a block column matrix with element-wise factors

Is it possible to factor this matrix $$\begin{bmatrix} x_{11} a_{11} & x_{11} a_{12} & x_{12} a_{11} & x_{12} a_{12} & \\ x_{21} a_{21} & x_{21} a_{22} & x_{22} a_{21} ...
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36 views

Linear Algebra Matrix Question solutions

Hi I was just wondering if an augmented matrix had no pivot positions, would the system have infinite solutions? Since it has no pivot positions that means, the columns must be filled with 0s and it ...
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2answers
144 views

Showing that $A=B+\alpha \cdot I$ is an invertible matrix

Let $B$ be a non-zero random $n\times n$ matrix generated using the matlab command $B=rand(n,n)$. I need to show that $A=B+\alpha \cdot I$ is an invertible matrix, where $\alpha=\|B\|_{\infty}$. I ...
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Find the rank of $A$ for all real $\lambda$.

Let $$A = \begin{bmatrix} {7 - \lambda } & { - 12} & 6 \\ {10} & { - 19 - \lambda } & {10} \\ {12} & { - 24} & {13 - \lambda } \\ \end{bmatrix}$$ Find ...
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34 views

The entries of a random matrix

I am confused about something. Whenever I create an $n\times n$ random matrix using Matlab (using the command $A=\mathrm{rand}(n,n)$), I get a square matrix whose entries are all between $0$ and $1$. ...
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2answers
43 views

Show this set forms an ellipse in R2

I'm trying to show that $\{ (x, y) : \|x\vec v +y \vec w \| = 1 \}$ in $\mathbb{R}^2$, with $\vec v,\vec w$ elements of a real inner product space, is the equation of an ellipse centered at $\vec 0$. ...
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65 views

Vector space and Dual space

I'm struggling with this problem: Let $V$ be a vector space over a field $F$ and let there be $l_1,l_2 \in V^*$. I need to show that if $l_1(x)l_2(x)=0$ for every $x \in V$ then at least one of ...
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1answer
40 views

Two different ways to write C(A)?

let $\mathrm A \in \Bbb R^{m\times n}$ I know that the three fundamental subspaces are: $\mathrm \ker(\mathrm A) = \{ x \in \Bbb R^n : \mathrm Ax = 0 \} = \{x\in \Bbb R^n : \langle ...
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2answers
57 views

linear systems?

I'm reading a book about linear systems, namely "Principles of LINEAR SYSTEMS and SIGNALS" by Lathi. The author states that the system to be linear must satisfies the superposition property which is ...
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1answer
52 views

Differentiability of linear least squares

Show that least-squares $\|y-X\beta\|^2$ is twice differentiable and has minimizer. I understand that the second derivative is $X'X$. Also it is a composition of linear function which is ...
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Reconstructing a matrix exponential from data

Suppose you have a matrix $A\in\mathbb{R}^{n\times n}$, and you are given vectors $x_i$, $y_i$ and scalars $t_i$ such that $y_i = e^{t_iA}x_i$. If you have $n$ of these vectors you should have ...
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1answer
72 views

Bilinear form on vector space

Does there exists a vector space $V$ and a bilinear form $w$ on $V\oplus V$ such that $w$ is not identically zero but $w (x,x) =0$ for every $x \in V$? My work is : if $M_2$ spanned by $\{(1,0,0,0), ...
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1answer
29 views

Finding a basis for any given subspace

I am wondering what the process is for finding a basis given any kind of subspace. In this example, I can picture just matrices with a 1 only in c, only in d, only in a, and only in a+b but I feel ...
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1answer
1k views

Proving that the eigenvalues of skew-Hermitian matrices are all pure imaginary

I would like some help on proving that the eigenvalues of skew-Hermitian matrices are all pure imaginary. I have gotten started on it, but am getting stuck. Attempt at proof: $Av=\lambda v \implies A ...
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1answer
154 views

Rotation of the Bloch Sphere

I was reading through the book "Quantum Computation and Quantum Information for Computer Scientists", and I got up to a problem about rotation matrices on the block sphere and I can't figure it out at ...
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173 views

Vector space of matrices, and subspaces

Show that the set of all real two-rowed square matrices form a vector space $X$. What is the zero vector? what is a basis? find $\dim X$. Give examples of subspaces of $X$. Do the symmetric matrices ...
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3answers
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Showing set linear independence

How do I show that the set $\{ e^x , ... ,e^{nx} \}$ is linearly independent? I tried using induction as the base case of $\{ e^x \}$ and even $\{ e^x, e^{2x} \}$ is easy, but I can't use the I.H. ...
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1answer
48 views

Prove/Disprove: The set linearly dependent

Let $T:V \rightarrow V$ such that $T^n = 0$ and let $A$ be the matrix representing $T$. Prove or Disprove: $\{I,A+I,(A+I)^2,...,(A+I)^n \}$ is linearly dependent. For the most questions I ...
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analytic solution to structured algebraic Riccati equation

In solving a model I have written down for a research paper, I am left with two Algebraic Riccati Equations, that is I need to solve for $X$ in the equation \begin{align*} X = A^\top (X + XB(R + ...
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1answer
224 views

Derivative of matrix inverse w.r.t. vector

I need to differentiate the inverse of the $K\times K$ symmetric matrix $A$ w.r.t some vector (that $A$ depends on). Is there a rule for this? In case I do the derivative w.r.t. to some scalar there's ...
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1answer
58 views

Linear Algebra: Find Matrix with Specified Kernel

I'm struggling with this problem: Find a matrix A whose kernel consists of all points in the plane $x + 2y + 3z$ = 0. I'm thinking the vector $[1, 2, 3]$ is perpendicular to any vector in the ...
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0answers
93 views

Prove image of basis is basis of vector space

With V and W being vector spaces, and T: V -> W being a linear transformation: c) Suppose B: (v_1, v_2, ..., v_n) is a basis for V and T is one-to-one and onto. Prove that T(B) = {T(v_1), T(v_2), ..., ...
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75 views

Inequality matrix norm

Let $A$ be an $n\times n$ random matrix $A=rand(n,n)$. Let $\alpha=max_{i,j}|a_{ij}|$ (i.e, $\alpha$ is the largest entry in $A$ in absolute value).I need to show that $\ \alpha < \| A \|_{2}$. ...
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31 views

Generalized Linear Least Squares

I've run across a problem which asks me to calculate a best fit line through data using a 'generalized linear least squares' approach where, instead of minimizing the residual: $\vec{r} = \vec{b} - ...
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1answer
2k views

Need help in understanding how to find an elementary matrix

I read this chapter in my book and thought I understood it, but I don't. I tried working a problem to test my understanding and I just don't know how to get started. Given the following matrices: ...
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1answer
26 views

Finding a basis. Technical issue.

I was asked to find a basis for the image of linear transformation. One easy way to do it is applying the linear map on $e_1,...e_n$, the standard basis. What do you do when $T(e_j) = 0$? Am I ...
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1answer
27 views

linear algebra-question on a linear transformation4

Let V be a vector space and W be any subspace. Then for given a vector space Y can you give a linear transformation such that W is the kernel of it?
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1answer
18 views

Criterions for 3x3 PD matrices

What are some sufficient conditions for a 3 by 3 Matrix to be positive (semi) definite? I ask because I need to know the convexity of $f(x,y,z)=-x^2-y^2-z^2+0.5xy$. I can compute the Hessian of $f$, ...
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3answers
125 views

The reason behind the name “Orthogonal transformation”.

An orthogonal transformation is a linear transformation such that $(Tx,Ty)=(x,y)$. Orthogonality is suggestive of perpendicularity. What might have been the reason for naming a distance preserving ...
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1answer
101 views

Determinant of a symmetric, positive semidefinite, sparse integer matrix

I'm looking for an algorithm that calculates the (log) determinant of a symmetric, positive semidefinite, sparse integer matrix. Does such an algorithm exist that can exploit both sparsity and ...
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1answer
36 views

Let $F$ be the set of infinite vectors $(a_0, a_1, …)$ satisfying $a_i + a_{i+1} = a_{i+2}$ for all $i \geq 0$. Show that $F$ is a vector space.

Here is my thinking. To show it's a vector space, we need to show that $(F, +)$ is an abelian group, and the other general axioms that make something a vector space. However, the way the operation is ...
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4answers
601 views

Proving $V$ is isomorphic to $W$ iff $\dim V=\dim W$

Let $V$ and $W$ be two finite vector spaces over $F$. Prove that $V$ is isomorphic to $W$ iff $\dim V=\dim W$ I think I got the general approach but I don't think it's rigorous enough. ...
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0answers
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properties of row reduction to explain $Ax - b = 0$ being true

How would i go about using properties of row reduction to explain why the equation $Ax - b = 0$ is true? I am not sure how to attack this. I know that $Ax=b$ where $b$ is a linear combination of the ...
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1answer
207 views

Eigenvalues and multiplication by diagonal matrices

I have $n \times n$ real matrices $A$ and $D$. $D$ is diagonal. Let's $v_i(A), \lambda_i(A)$ be a couple of eigenvectors-eigenvalues of $A$. What relationships there exists between $v_i(B), ...
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3answers
97 views

Proving a set of functionals is independent.

A functional is a linear transformation from an $n$-dimensional vector space $V$ to its scalar field $F$. I need to prove that a set of functionals $\{ f_1,....,f_n \}$ is independent (in the linear ...
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1answer
51 views

Short question about 'multilinearity' of determinant

We know that this is true and I think it's called multilinearilty of a determinant: $$\begin{vmatrix} z & x & c \\ a+b & a+b & a+b \\ q & w & e \\ \end{vmatrix} = ...
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Mathematical standard term for a function of (different) operator arguments

In quantum mechanics, one often considers functions of linear operators, like $$f(A,B) = A\cdot B + e^A \cdot B^2$$ where $A,B$ are linear operators. In physics this often causes confusions, as some ...
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Finding projections

I'm not completely sure about this kind of question: $$\begin{align}V&=\operatorname{span}\{(2,2,2,1),(1,0,1,1),(1,4,1,-1)\}\\ ...
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1answer
606 views

how to show a direct sum of two subspaces

Let $U,V$ two subspaces of $\mathbb{R}^n$, such that: $U = \{x \in \mathbb R^n | x_1+x_2+...+x_n=0\}$ and $V=\{x\in V |x_1=x_2=...=x_n\}$ Show that the sum of the two subspaces is a direct sum. ...
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769 views

determine whether the equation $Ax = b$ is consistent for every $b$ in $\mathbb R^m$

I have two problems, the first one is the following matrix: $$\begin{bmatrix}1 & 0\\ -2 & 1\end{bmatrix}$$ where the RREF is $$\begin{bmatrix}1&0\\0&1\end{bmatrix}$$ and where the ...
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1answer
56 views

(homework) Proving that a product of two matrices is in RREF

From Anton's Elementary Linear Algebra 8e Chapter 1.5, #14: Prove that if $A$ is an $m\times n$ matrix, there is an invertible matrix $C$ such that $CA$ is in reduced row-echelon form. I know ...
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875 views

What are the best texts on undergraduate linear algebra?

I have recently finished a course in 'elementary linear algebra,' which entails basic systems of linear equations, in-depth study on matrices, the basics of vector space, inner product spaces, linear ...
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35 views

Dimension Theorem modification

The Dimension Theorem says $$ \dim(U+W) = \dim(U) + \dim(W) - \dim(U \cap W) $$ The proof of this theorem uses the bases of $U$, $W$, and $U\cap W$. Is it possible to prove this theorem with just ...
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1answer
25 views

Is this some sort of directional derivative problem I have here?

Let $V = y^2U_{1} - xU_{3}$. Also, let $f = xy$ and $g = z^3$ Compute $V[f]$ and $V[g]$. Now $U_{1} = (1,0,0)$ and $U_{3} = (0,0,1)$ Now in my notes, $V_{p}[f] = \displaystyle\frac{d}{dt}(f(p + ...