Questions tagged [linear-algebra]

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 concerning matrices, use the (matrices) tag. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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2answers
34 views

Inner product of infinite dimensions

I want to know the reason why we can use integral for inner product in infinite dimensional space. i.e.: Why the innerproduct, which in finite dimension is $\sum_{i}f(x_{i})g(x_{i})$, in infinite ...
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1answer
52 views

Given a matrix $A\in M_3(Z_{11})$, calculate $\det(3A)^{-1}$ given $\det A=4$

$(3A)^{-1} = \frac{1}{3}A^{-1}$ So that, $\det(3A)^{-1} = (\frac{1}{3})^3\det(A^{-1}) = \frac{1}{27}*\frac{1}{4} = \frac{1}{9}$ Am I correct?
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1answer
22 views

Taking positive part commutes with conjugating with $Y\geq 0$ on hermitian matrices?

Let $X,Y\in\mathbb C^{n\times n}$ be hermitian and $Y$ positive semi-definite. Does $$ (YXY)^+=YX^+Y $$ hold, where $(\cdot)^+$ denotes the positive part of the respective hermitian matrix (i.e. if $A=...
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1answer
23 views

Prove that if vi is a basis of the column space of A then f(vi) is a basis of the column space of f(A)

I met a problem from the liner algebra test of our school. The question is: Consider the following function $f$ that takes any matrix $A$ as argument and returns a matrix $$f\left(A\right)=\frac1{\...
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2answers
91 views

There does not exist a onto ring homomorphism from $M_{n+1 \times n+1}(\mathbb F) \to M_{n \times n}(\mathbb F) $ for any field $\mathbb F.$

For a positive integer $n$, I have to show that there does not exist a onto ring homomorphism from $M_{n+1 \times n+1}(\mathbb F) \to M_{n \times n}(\mathbb F) $ for any field $\mathbb F.$ If it ...
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2answers
41 views

Convert Rotation matrix to Euler angles $~zyz~ (y$ convention$)$ analytically.

The rotation matrix of Euler angle $ZYZ$ is: $$ R_{z1}=\left[\cos(\psi),\sin(\psi),0;-\sin(\psi),\cos(\psi),0;0,0,1 \right]; $$ $$ R_y=[\cos(\theta),0,-\sin(\theta);0,1,0;\sin(\theta),0,\cos(\theta)]; ...
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2answers
24 views

Finding the points that line on a plane

Let $P$ denote the plane given by the point-normal equation: $0 = (1,2,−1)·((x,y,z)−(1,1,1))$ How do I find the points $(x, y, z)$ that lie on the plane $P$?
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1answer
36 views

Find a and b such that (1,1,1) = a(1,-2,1) + b(2,-1,-1) where a,b belong to R [closed]

Find $a$ and $b$ such that $(1,1,1) = a(1,-2,1) + b(2,-1,-1)$, where $a,b$ belong to $\mathbb{R}$. I was not able to find a solution for this. Can someone help?
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5answers
292 views

Demystifying the Determinant

Preface: I’ve read a few of the posts on here about the determinant but none seem to put to rest the the questions I have, so I’ve decided to ask outright. Question: In general, I would like to just ...
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1answer
28 views

Lax-Milgram Lemma, Alternative Proof in Finite Dimensional Case

I am asking myself, if we would weaken the assumptions of the Lax-Milgram Lemma to the finite dimensional case Lax Milgram Lemma Let ($V$, $(\cdot, \cdot$, $\Vert \cdot \Vert$) be a (real) Hilbert ...
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1answer
42 views

Sum of determinants of block submatrices

I have a $2n \times 2n$ matrix, $M$. I view it a block matrix, of $n^2$ blocks, each of shape $2\times 2$. Computing the determinant of $M$ is easy by conventional methods. I could also look at ...
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3answers
56 views

Show that the vectors $v_1 = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $v_2 = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ are linearly independent…

Show that the vectors $v_1 = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $v_2 = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ are linearly independent, and find the unique coefficients $x$ with \begin{equation*}...
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1answer
73 views

Lower bound $\operatorname{Tr}(MB)$ where $M>0$ and $B$ is arbitrary

Let $M, B$ be complex and square matrices. $M$ is positive definite (pd) and $B$ is arbitrary. If this is relevant, there is an afore-knowledge that $\operatorname{Tr}(MB)\ \ge \ 0$. Please help to ...
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2answers
113 views

Computing $(A\otimes I + I \otimes A)^{-1} \text{vec}B$

Suppose I have two positive semi-definite $n$-by-$n$ matrices $A$, $B$ and an $n$-by-$n$ identity matrix $I$, and I'm looking for a way to compute, approximate or bound the following quantity: $$(A\...
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1answer
38 views

Let $W = \operatorname{span}\{\mathbf{w}_1\}$, where… [closed]

Just checking to make sure this is right!!! Let $W = \operatorname{span}\{\mathbf{w}_1\}$, where \begin{equation*} \mathbf{w}_1 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}. \end{equation*} 1. Find a ...
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0answers
16 views

Two definitions for a quadratic form $Q$ to be nondegenerate

Let $(V,Q)$ be a pair of vector space $V$ over $k$ of characteristic not equal to $2$ and a quadratic form $Q$ over $V$. Put a symmetric bilinear form: $$x.y:=\frac{1}{2}(Q(x+y)-Q(x)-Q(y)).$$ The ...
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0answers
16 views

Prove existence of a basis given a matrix and a linear transformation describing matrix (Axler 3, #5)

I have found an answer here, but I found such answer to be unclear/unsatisfactory to me. So, I wrote out my own rationale and would like feedback on the proof. Problem Statement Suppose $w_1,...,w_n$...
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0answers
24 views

Bound of eigenvalues of product of two positive definite matrices [closed]

Suppose I have two positive definite matrices A and B, Eigenvalues of A are bounded between u, v Eigenvalues of B are bounded between s, t All real numbers u,v,s,t are in range [0, 1]. I can infer ...
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2answers
29 views

Span of Null Space of A

Apologies for the brief nature of this question, but it is something that I don't think was clarified in a previous post on this topic - Finding a spanning set for a null space. When we say that the ...
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0answers
20 views

how to calculate wire per metric ton by using maths? [closed]

I have $4$ questions $1.$ If a $1.2$ mm thick wire costs $\$6$ per meter per kg then how much will be the of $1000$ kg? $2.$ If a $1.5$ mm thick wire is $\$7.98875$ per meter then how much will be ...
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1answer
47 views

Why does such an orthogonal vector exist?

Suppose $W$ and $V$ are $k$-dimensional subspaces of $\mathbb{R}^m$ for some $m \geq k$. Let $V$ have orthonormal basis $\mathbf{v_1}, \mathbf{v_2}, \ldots, \mathbf{v_k}$. Then is it possible to ...
2
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1answer
60 views

Vector subderivatives and “simple algebra” which turn out not to be so simple

In Friedman, Hastie and Simon (2013) an algorithm is proposed for a group-LASSO penalized regression possibly involving many variables. The problem is as follows: $\underset{\beta}{min}\{ \frac{1}{2}|...
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1answer
44 views

Relation between definition of determinant and UFD

I've asked the question about inductively displaying Vandermonde determinant as a product of polynomials and one of the answers I got had an important piece of information within: From the ...
2
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3answers
61 views

Maximum number of pairwise linearly independent vectors

Consider vectors $v_1,\dots,v_n\in\mathbb{R}^d$. My question is: What is the maximum number of such vectors, that are pairwise linearly independent? Clearly, if we remove the word pairwise the ...
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1answer
49 views

SVD of a specific upper triangular matrix

Given a matrix $A$: $$ A = \begin{pmatrix} a_{11} & 0 & a_{13} \\ 0 & a_{22} & a_{23} \\ 0 & 0 & 1 \\ \end{pmatrix} $$ All the parameters different from $0$ are strictly ...
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1answer
25 views

infinite order automorphism on torus

Let $T= \mathbb R^n/\Gamma$ a torus, where $\Gamma$ is the standard lattice in $\mathbb R^n$. A matrix $A\in SL(n,\mathbb Z)$ induces an automorphism $A_T$ on $T$. I want to calculate the order of $...
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1answer
21 views

Orthogonal Complement and Subspace

If the vectors $a_1,...,a_k$ generate the subspace $V$ of $\mathbb{R}^n$, and $x\in\mathbb{R}^n$ is orthogonal to each of these vectors, show that $x\in V^\perp$. My attempt: Let $y\in V$. Then, we ...
2
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1answer
28 views

Right-inverse approximation in Frobenius Norm

Let $A \in \mathbb{R}^{m\times n}$, with $m \geq n$, be a matrix of rank $r$, and suppose we have a SVD decomposition $A = U\Sigma V^t$. We define the pseudo-inverse of $A$ as $A^{\dagger} := V\...
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1answer
27 views

Finding a subspace of image of linear transformation

The question is: Consider the linear transformation $L: \mathbb{R}^3 \to \mathbb{R}^3 : (x,y,z) \mapsto (2x+y-z,y-2z,-2x-z)$. When $U = \text{Span}\{(0,0,1),(1,1,1)\} \subset \mathbb{R}^3$ is a ...
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2answers
58 views

Proof verification: if $W_1 \subseteq W_2$ then $\dim(W_1) \le \dim(W_2)$

Let $W_1$ and $W_2$ be subspaces of vector space $V$. Prove that If $W_1 \subseteq W_2$ then $\dim(W_1) \le \dim(W_2)$. My proof: Let $v_1, ...,v_n$ be base vectors of vector space $W_1$ and $W_1 \...
2
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0answers
21 views

Why are arithmetical operations on during row reduction counted in this way?

I am reading the section on counting arithmetic operations in Strang's book. It says: [The] operations are of two kinds. We divide by the pivot to find out what multiple (say $\ell$) of the pivot ...
3
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6answers
114 views

Homogeneous Equations and Linear Algebra

For the past few months I have been learning about the fundamental concepts of linear algebra in order to make my life as an aspiring game developer easier. Whilst working on the topic of null space, ...
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0answers
33 views

How to find projection operator from a matrix without knowing basis?

I have a 6 by 6 matrix $A$. I have calculated its eigenvalues and eigen function. Now I need to take projection on a subspace (upper left 3 by 3 block matrix). How can I do that? Projection operator ...
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1answer
26 views

What are the necessary conditions for a matrix to have a complete set of orthogonal eigenvectors?

From my lecture notes, I learned that for a $N\times N$ real symmetric matrix $\mathbf{A}$, it is known that it has a complete set of $N$ orthogonal eigenvectors $\hat{e}^{k}$, with $k=1 \ldots N$ ...
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2answers
82 views

Find a general solution for $y'=\frac{3x+4y+2}{2x+y+3}$

$$\frac{dy}{dx}=\frac{3x+4y+2}{2x+y+3}$$ I tried to solve this problem as following, like 'exact form problem', $$(2x+y+3)dy - (3x+4y+2)dx = 0$$ This equation is not exact yet, so i tried to find ...
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2answers
30 views

The Optimal Rotation Around a Fixed Axis

So I have two sets of points, $P$ and $Q$, in the 3D Euclidian space. Set $P$ has $N$ points and set $Q$ has $M$ points. It is possible that $N=M$ but not necessarily. Now I define quantity $S$ as ...
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2answers
101 views

Proof that the $\dim (U+W)= \dim U + \dim W -\dim (U\cap W)$

The Book is Bosch Linear Algebra. I have difficulties to understand how one deduces this statement from prior statements proved earlier: Statement already proved: Definition 1. Let $U_1,....,U_r$ ...
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0answers
38 views

Generation of a basis of an infinite dimensional space using a finite number of vectors

Suppose the set $\mathscr{H} = \{\phi_i\}_{i=1}^{\infty}$ is a complete orthogonal set (of functions) with a given inner product operation. Let $\mathscr{H}_N = \{\phi_i\}_{i=1}^{N}$, i.e. the first $...
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0answers
24 views

Reference Request: Algebraic and/or geometrical study on closed flat space

I am curious about studying the following subset $X$ of $\mathbb{R}^n$: $$X:=\{(x_1,\cdots,x_n)\in\mathbb{R}^n|x_i\ge 0\ \mathrm{for}\ \forall i,\sum_{i=1}^nx_i=1\}.$$ I can see that $X$ is ...
1
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1answer
58 views

Let $W = \textit{span}\{\mathbf{w}_1,\mathbf{w}_2,\mathbf{w}_3\}$…

Let $W = \text{span}\{\mathbf{w}_1,\mathbf{w}_2,\mathbf{w}_3\}$, where \begin{equation*} \mathbf{w}_1 = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}, \quad \mathbf{w}_2 = \begin{pmatrix} 1 \\ 0 \\ 1 \end{...
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2answers
52 views

Basis (linear algebra) - ordering of the vectors?

Looking at Wikipedia Basis (linear algebra), it seems that in English a basis is a set of vectors that may be ordered, with ordering not being mandatory. This creates issues when dealing with ...
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1answer
30 views

Showing exterior product is skew-commutative

I am learning about the exterior product and wish to show the skew commutative property. Here is my work: I am confused on how to handle the notation of $(\omega^k \wedge \omega^l)$ as at the end I ...
2
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4answers
47 views

Consider the matrix form $A\mathbf{x} = \mathbf{b}, \ A\in \mathbb{R}^{m\times n}$.

Consider the matrix form \begin{equation*} A\mathbf{x} = \mathbf{b}, \quad A\in \mathbb{R}^{m\times n} \end{equation*} of a linear system of $m$ equations in $n$ unknowns $\mathbf{x} = (x_1,\ldots,x_n)...
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1answer
31 views

Is the number of ordered and unordered bases of a vector space is same?

(1) Is the number of ordered and unordered bases of a vector space is same? (2) Upto independent is same or not ? My approach: (1) Let $B=\{v_1, v_2, \cdots, v_n \}$ be an ordered basis. Then ...
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0answers
25 views

Condition on matrix to give a cylinder

The question is to put a condition on the 2x2 matrix $H$ which is in the form $$ \left( \begin{matrix} z & w\\ -\bar{w} & \bar{z}\\ \end{matrix} \right) $$ where z and w are complex numbers....
2
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0answers
35 views

closure of intersection under projection

Let $Z=\{(x,y)\in R^d\times R^l: a\leq Ax\leq b, y=Bx\}$ where $A,B$ are matrices and $a, b$ are vectors in $R^m$, with $a_i\leq b_i$, for all $i=1,...,m$. Then $Z=Z_1\cap Z_2$ where $Z_1=\{(x,y): ...
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1answer
35 views

No bijection between quadratic forms and symmetric bilinear forms when the field is of characteristic 2

Let $V$ be a vector space over $k$, a field of characteristic $2$, I wonder how to show that in general there is no bijection between quadratic forms and symmetric bilinear forms. I understand that ...
2
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1answer
27 views

Finding the Jordan form given nullities

The question: "Let $B$ be a $10 \times 10$ matrix and let $\lambda$ be a scalar. Suppose it is known that $$ \text{nullity}(B - \lambda I) = 5, \\ \text{nullity}(B - \lambda I)^2 = 8, \\ \text{...
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1answer
19 views

Polynomial Space Subspace

Let $S$ be contained in $P_n([-5,5])$ such that $S=\{f(x) \in P_n[-5,5]$ | $f'(-1)=0, f''(1)=0\}$ Check if S is a subspace of $P_n(-5,5)$ I have absolutely no idea how to go about this, (topic is ...
0
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1answer
17 views

$S+T$ is direct iff $A,B$ are disjoint for all bases $A$ of $S$ and $B$ of $T$

Show that for two subspaces $S,T$ of a vector space $V$, $S+T$ is direct iff $A,B$ are disjoint for all bases $A$ of $S$ and $B$ of $T$. I am not so familiar with direct sums but all I know that $S+T$...