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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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3answers
71 views

How to Solve a AU=B System when Determinant of A=0

Let $$A=\begin{pmatrix} 1 & 5 & -1\\ -2 & -10 & 5\\ -2 & -10 & -1\end{pmatrix}$$ and $$B=\begin{pmatrix} 0 \\ 15 \\ -15\end{pmatrix}.$$ To find a vector $U$ such that $A$ ...
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1answer
13 views

Stability of linear autonomous systems

Let $x'=Ax, A\in \mathbb{R^{n,n}}$, be a linear autonomous system. Denote by $\{ \lambda_j\}$ the set of the eigenvalues of $A$. I want to study its stability. There is this following fact that im ...
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1answer
31 views

Big-o notation confused

In error analysis we say the error is of order $\epsilon$ if the error is less than or equal constant multiple of $\epsilon$. The question is: what is the benefit we have if we can multiply by any ...
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0answers
8 views

Convergence of unit vector in dir. of $M^{k}*v$ to the principal eigenvector of M when $k\to \infty$ and M is symmetric

It is a standard fact that a square matrix $M$ of dimension $n$ has at most $n$ distinct eigenvalues, each of them a real number, and the sum of their multiplicities is exactly $n$. We will denote ...
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1answer
27 views

Proving Euclidian Norm squared is equivalent to transpose times matrix for x in R^n

Apologies if this has been answered already but I can't seem to find an answer that I think answers my question (or at least one I understand). Anyways the question is, ...
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2answers
27 views

Invariant Subspaces and Orthogonality

I'm struggling with a question on invariant subspaces. If someone could possibly help me out here that would be great. The question is as follows: Let $A ∈ M$ where $M$ is the set of $n \times n$ ...
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0answers
42 views

When diagonalizing a matrix does the order of the eigenvectors matter?

Say I find the eigenvalues of some matrix $A$ and then use kernel computation to find an eigenbasis for each eigenvalue. Then $B = S^{-1}AS$, where $S$ is composed of the eigenbasis of each of the ...
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2answers
49 views

What is the Geometric meaning of vector norm in Rn n>3

My question is related to the length of the vector , Sorry it may seem stupid for you as i come from engineering background not mathematics background For Vectors up to 3 dimensions (can be ...
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0answers
22 views

T/F: If a subspace S has a basis B with 2 elements then S is a plane.

I believe the answer is yes, because even if the basis is in $R$2, then the basis would represent the xy-plane. For $R$3 and beyond, two elements of a basis should always form a plane Is my ...
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0answers
18 views

Generalize “matrix multiplication” with an function/operator?

Newbie question. No idea if this make sense or where to ask. Matrix multiplication is defined as, “If A is an n×m matrix and B is an m×p matrix, their matrix product AB is an n×p matrix, in which the ...
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1answer
36 views

What rotations are performed to produce this output on a Tesseract?

I'm writing a program that projects a tesseract in 3D on a 2D environment and I want to reproduce the rotation of this gif but I'm having difficulty grasping what rotations before projection I need to ...
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2answers
32 views

Does double projection preserve lines?

Let $ V $ and $ W $ be two subspaces of Euclidean space $ \mathbb{R}^n $ such that $ \dim{V}=\dim{W}, x\in V $. Is it true, that $ \mathrm{Pr}_V(\mathrm{Pr}_W(x)) = c \cdot x $ for some $ c\in\mathbb{...
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1answer
22 views

The kernel of $\bigwedge^r(T): \bigwedge^r(W)\to \bigwedge^r(V)$?

The full problem is: Given $T: W\to V$, a linear transformation of $F$-vector spaces, such that $\text{ker} T = 0$, show that the kernel of $\bigwedge^r(T): \bigwedge^r(W)\to \bigwedge^r(V)$ is ...
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2answers
38 views

How to correctly write the set of all vectors of the form $(a,b,c)$, where $a+2b-c=0$

I'm not 100% sure if this is where this should go, but my LA prof wants us to practice writing more mathematically using correct notation. this is one of our homework problems and I wrote down the ...
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1answer
17 views

Strong Duality: If Dual is optimal then primal is optimal

Strong duality states that if the Primal has an optimal solution then the Dual has an optimal solution. Is the converse of this statement true? To me it would seem intuitive that it is, but I just ...
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1answer
52 views

Let $A,B \in (X \to X)$ be linear operations such that $A^2=B^2$ and $ KerA \cap KerB= {\{0}\}$. Show that

Let $A,B \in (X \to X)$ such that $A^2=B^2$ and $ KerA \cap KerB= {\{0}\}$. Show that $a) A(\mathrm K\mathrm e\mathrm rB) \subseteq \mathrm K\mathrm e\mathrm rA$ $b) \ \mathrm d\mathrm i \mathrm mA(...
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1answer
28 views

Find An Eigenvalue & Eigenvector from a given Eigenvalue and eigenvector

I was given a 3x3 matrix and I was given a complex Eigenvalue of -2+6i. They also gave me the eigenvector [-6-2i] [1] [9i]. I have to find another eigenvector and eigenvalue that isn't given. I ...
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2answers
32 views

Ellipsoid and linear transformation

I get confused in my algebra about this simple problem. The equation of a 3D centred ellipsoid can be written on a compact form as $$x^{T}A x=1$$ with $x\in \mathbb{R}^{3}$ et $A$ a matrix that ...
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2answers
18 views

Polynomial of Linear Operator

Let $L$ denote a linear operator and $v\in V$. Does the expression $$c_0L^nv + c_1L^{n-1}v + \cdots + c_{n-1}L^1v + c_nL^0v = 0$$ has a special name and what properties are known? For example, I know ...
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0answers
24 views

Geometry of Solution Set and Matrix

Hello all. If wrote this system as a matrix, I would recognize it as a 3x5 matrix, but what I am not confident on is the fact that there are four variables, with constants on the other side. My gut ...
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1answer
30 views

Confused about finding non-brute force way to solve for matrix to the 2019th power

I am attempting to solve this problem, it has four parts. I solved part a (a trivial matrix problem), but the next three parts appear to be a bit confusing to me. I just would like some help getting ...
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2answers
18 views

What do you mean by scaling a vector in a particular direction?

Find a 2 × 2 matrix $A$ such that the linear transformation $T : R^2 → R^2$ defined by $T(~u) =A~u$ geometrically scales by a factor of 2 in the direction of the vector $\left(1,2\right)^T$ and scales ...
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1answer
23 views

What is the total number of distinct $m\times n$ matrices in row canonical form using only $0$s and $1$s?

Suppose that $A$ is an $m \times n$ matrix over a field $F$. What is the total number $N$ of the distinct matrices in row-reduced echelon form that are row equivalent to $A$ and that only have entires ...
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1answer
12 views

Showing $\Omega = PP'$ implies $P^{-1}\Omega(P')^{-1} = I$, where $\Omega$ [on hold]

Similar as the title, how to prove that $\Omega = PP'$ implies $P^{-1}\Omega(P')^{-1} = I$, where $\Omega$ is a symmetric and positive definite matrix.
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2answers
48 views

Find two vectors X, Y [on hold]

Let $$C=\begin{pmatrix} 1 & 2 & 3 \\ -1 & 1 & 1 \\ 1 & 0 & 1 \\ \end{pmatrix}.$$ Find two vectors $X, Y \in R^3$ such that $X^TCY \neq Y^TCX$.
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2answers
61 views

When do two vector spaces with same dimension have the same basis?

I was thinking of this case: two vector spaces - $\mathbb{R}^2$ over $\mathbb{R}$ and $\mathbb{C}^2$ over $\mathbb{C}$. Every basis in the first vector space is also a basis for the second vector ...
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0answers
21 views

Gram-Schmidt orthogonalization process with specific dot product

I have three vectors $$v_1=(1,1,1)^T$$ $$v_2=(1,1,0)^T$$ $$v_3=(1,0,0)^T$$ and special dot product definition $$(\overline{(x_1,x_2,x_3)},\overline{(y_1,y_2,y_3)})=2x_1y_2+x_1y_1+2x_2y_1+x_3y_3$$ I ...
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1answer
19 views

Find $A(v_1+v_2)$ and $A(3v_1)$ given eigenvectors and eigenvalues

If $v_1=[-1;5]$ and $v_2=[-3;5]$ are eigenvectors of a matrix $A$ corresponding to the eigenvalues $\lambda_1=-1$ and $\lambda_2=1$, find $A(v_1+v_2)$ and $A(3v_1).$ I managed to find $A,$ ...
2
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1answer
42 views

Finding the basis of a subgroup

Let $A$ be a subgroup of $\mathbb{Z}^n$ of index $3$. Prove that there exists a basis $v_1,...,v_n$ of $\mathbb{Z}^n$ such that $A$ is generated by $v_1,...,v_{n-1},3v_n$. My attempt: Let $B_1 = \{...
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1answer
21 views

Rewriting product of special rank one updates as a low rank update

I'm trying to improve the speed of the following iteration to calculate $s_k$: $$B_k^{-1} = \Bigg( I + \frac{s_{k}s_{k-1}^T}{||s_{k-1}||^2}\Bigg)...\Bigg(I+ \frac{s_1s_0^T}{||s_0||^2}\Bigg) B_0^{-1}\\...
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0answers
20 views

Dimension of a subspace of a Vector space [on hold]

Let $𝑇 \in 𝑀_{𝑚\times 𝑛}(\mathbb{R})$. Let $𝑉$ be the subspace of $ 𝑀_{𝑛\times 𝑝}(\mathbb{R})$ defined by $𝑉 = \{𝑋 \in 𝑀_{𝑛\times 𝑝}(\mathbb{R}) ∶ 𝑇𝑋 = 0\}$. Then the dimension of $𝑉...
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4answers
36 views

How To Find The Unit Eigenvectors

I have the matrix $$\begin{pmatrix}3&-9\\-9&27\end{pmatrix}.$$ I found the eigenvalues of $0$ and $30.$ However, when I try to plug in $0$ for the Eigenvalues and row reduce, I get $0,0$ as my ...
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0answers
19 views

Confusion with the QR decomposition

I have some trouble to understand this: According to the factorization QR, Given a matrix $A\in\mathbb{R}^{n\times p}$, with $rank=min(n,p)$, there exists an orthogonal matrix $Q\in\mathbb{R}^{n\...
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2answers
18 views

2 vector of equal length given how to find vector about which one vector reflected to that other?

Let $ w\in \mathbb R^n$ be vector of length $1$. $U$ is orthogonal space $w^\perp $ The reflection $r_w $ about $U$ is defined as follows if $v=cw+u$ , $u\in U$ then $r_w(v)=-cw+u$ Let $ u ,v$ be ...
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0answers
11 views

Eigenvalues of real part of positive definite hermitian matrix

Lets say we have the $n\times n$ positive definite hermitian matrix $\mathbf{A}$. Is there any clear relation between the Eigenvalues of $\mathbf{A}$ and the Eigenvalues of its real part $\mathbf{B}=\...
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0answers
29 views

How To Use A Given Gauss Elimination To Find A Determinant

I was given an equation for Gauss elimination and am unsure of how to solve it with the given equations. I don't know if I'm multiplying wrong. But, they gave the elementary matrixes throughout the ...
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2answers
44 views

Invertibility of $I + AB(x)$

I am dealing with a matrix $$I + AB(x)$$ where $A, B(x)$ are square $n\times n$ real matrices and $x$ is a real variable. I want to find the values of $x$ for which this matrix is singular (and then ...
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1answer
26 views

Vector space of recursive sequences

I never faced this kind of vector space, so i'm a little insecure with how to solve this. Question: Let $a,b\in\mathbb{C}$ be non vanishing complex numbers. We consider the space of sequences defined ...
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0answers
36 views
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1answer
11 views

If colums of an $n\times n$ matrice are linearly independent then columns of transpose of this matrice are linearly independent.

My question is a very short: ''If colums of an $n\times n$ matrice are linearly independent then columns of transpose of this matrice are linearly independent'' Is this true? Can you explain? Thanks....
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1answer
47 views

Property of some determinant [on hold]

Prove that $$ \begin{vmatrix} 1+a^2-b^2 & 2b & -2b \\ 2ab & 1-a^2+b^2 & 2a\\ 2b & -2a^2 & 1-a^2-b^2\\ \end{vmatrix} $$ is a perfect cube.
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2answers
66 views

Is $0$ an eigen value of $A$

Consider the following matrix: $A=\begin{bmatrix} 9&1&1&1&1&1&2&2\\1& 9&1&1&1&1&2&2\\1&1&9&1&1&1&2&2\\1&1&...
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0answers
11 views

Multiplying A matrix by series of matrixes, At thr same time

B is a matrix. E is an elementary matrix. A is a matrix. All of them have appropriate sizes. B=$E_1E_2E_3A$ What kind of order Should I follow in multiplying these matrices?
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1answer
68 views

Groups generated by a subset

Let $G$ a group, $S,T \subseteq G$ and $\langle S \rangle, \langle T \rangle$ the subgroups generated by $S$ and $T$. Is it true or false that $$\langle \langle S \rangle \cup \langle T \rangle\rangle ...
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1answer
31 views

Dot product for orthonormal basis

I want ask which must be the dot product for vectors (1,1,0), (1,0,1), (0,1,1), so they can form a orthonormalbasis.
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1answer
20 views

How to show that the matrix $R^TCR$ is diagonal if $R$ is a rotation matrix related to $C$ in a specific way?

I have two matrices: $C=\begin{bmatrix}c_{11}&c_{12}\\c_{21}&c_{22}\end{bmatrix}$ and $R=\begin{bmatrix}\cos\theta &-\sin\theta\\\sin\theta & \cos\theta\end{bmatrix}$. I would like to ...
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0answers
20 views

Understanding the Power Method to find eigenvalues

For $A=\sum_i\sigma_i u_iv_i^T$, let $B=A^TA=\sum_i\sigma_i^2 v_iv_i^T$. Then $$B^k=\sum_i\sigma_i^{2k}v_iv_i^T$$ If $\sigma_1>\sigma_2$, then $B^k\to \sigma_1^{2k}v_1 v_1^T$. I don't know why a ...
0
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1answer
20 views

Struggling to prove that the elements of an inverse matrix satisfy a certain equation.

We have three vectors $\vec{e_1},\vec{e_2},\vec{e_3}$ that are not necessarily orthogonal or normalised, but do form a basis. We also have a matrix $G$ with elements $G_{ij}=\vec{e_i} \ . \vec{e_j}$,...
0
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1answer
21 views

Getting different answers for same problem on divergence and curl.

Given that $\vec{a}$ is a constant vector and $\vec{r}$ is a position vector. We are asked to prove the following: $$\nabla\times(\vec{a}\times\vec{r})=2\vec{a}$$ I tried two ways. Could prove it ...
2
votes
2answers
74 views

How to compute the smallest eigenvalue efficiently? [on hold]

$A$ is a $m \times m$ symmetric PSD matrix whose top $n$ eigenvalues are equal to $1$ and whose remaining $(m-n)$ eigenvalues are zero. Here, $n \ll m$. Let $D$ be a diagonal matrix with all diagonal ...