Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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298 views

Normal distributed rotation matrix in 3D

How can I compute normally distributed 3D rotation matrices with Mathematica? For 2D matrices I would sample a normal distributed angle and directly create a rotation matrix with: ...
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122 views

Solve covariance matrix of multivariate gaussian [duplicate]

This is a practical, and basic question. I have a multivariate Gaussian in $M$ dimensions with center $\mu$ (known, lets assume $0$) and some points $p$ where I have the value of $$ \ln(L)= ...
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101 views

Prove that $C^n(\mathbb{R})$ is a subspace using induction.

Let $V$ be the set of all functions $f:\mathbb{R}\to\mathbb{R}$. Prove by induction that $C^n(\mathbb{R})$ is a subspace of $V$. I feel that this could be shown directly without much issue using the ...
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81 views

How to solve matrix differentiation

How to solve: $${\frac{\partial}{\partial{\vec{\mu}^T}}\left\{\frac{1}{a}\sum^n_{i=1}z_i(\vec{y_i}-\vec{\mu})^TB^{-1}\right\}}$$ where $a$ and $z_i$ are elements, $\vec{y_i}$ and $\vec{\mu}$ are ...
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92 views

Significance of order in linear algebra

This question has been bothering me for some time now. All theorems in my linear algebra notes use ordered lists of vectors. For example consider the theorem here: A spanning list of vectors may ...
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58 views

When polynomial is power

$P(x)$ ia a polynomial with real coefficients, and $k>1$ is an integer. For any $n\in\Bbb Z$, we have $P(n)=m^k$ for some $m\in\Bbb Z$. Show that there exists a real coefficients polynomial ...
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79 views

A is symmetric iff A=P-Q, where P,Q are positive definite matrices

Show that an $n \times n$ real matrix $A$ is symmetric iff $A$ can be written as $$A=P-Q$$ where $P$ and $Q$ are some $n \times n$ positive definite matrices. Can there be anything said similarly ...
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326 views

Making non-singular matrices singular

What is the minimum value of $k$ such that every non-singular $n\times n$ real matrices can be made singular by switching EXACTLY $k$ entries with ZERO ?
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142 views

Subspace spanned by eigenvectors is a subalgebra

Let $L$ be a Lie algebra over an algebraically closed field and let $x\in L$. Prove that the subspace of $L$ spanned by the eigenvectors of $\operatorname{ad}x$ is a subalgebra. Suppose the ...
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51 views

Stumped with Matrices

(a) How do we find $A^{-1}$? (b) If $XA=B$, how do we use (a) to find $X$? Any guidance would be much appreciated!
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Changing a simplex grid to an orthogonal grid.

Well I'm on my way in learning noise, a computer algorithm that's used to create real life structures and textures, etc. The noise I'm trying to learn is Simplex Noise, I already have it in the ...
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38 views

Fourier transform of a sequence by a matrix

Let $n$ be a positive integer and $H$ the Hilbert space $\ell^2(\mathbb Z^n,\mathbb C^n)$. For $u\in H$, denote by $\mathcal{F}(u)$ the Fourier transform of $u$, defined by $\displaystyle ...
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351 views

If the determinant and the trace of a matrix are the same, then can we show they have the same eigenvalues?

If $\det A = \det B$ and $\operatorname{tr}A=\operatorname{tr} B$, then can we show $A$ and $B$ have the same eigenvalues?
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Finding a basis for $U=\{A\in\mathbb{M}_{22}\mid A^T=-A\}$.

Find a basis for $U=\{A\in\mathbb{M}_{22}\mid A^T=-A\}$. $\mathbb{M}_{22}$ denotes the set of all $2 \times 2$ matrices. This question appeared on an examination I wrote yesterday. Does a basis ...
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1answer
40 views

Direct sum and similarity question

Let $A,C$ be $n\times n$ matrices and $B,D$ be $m\times m$ matrices. Assume $A\oplus B$ is similar to $C\oplus D$. Then, are $A&C$ similar and $B&D$ similar respectively? If they are, how ...
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286 views

How to find the degrees between 2 vectors when I have $\arccos$ just in radian mode?

I'm trying to write in java a function which finds the angles, in degrees, between 2 vectors, according to the follow equation - $$\cos{\theta} = \frac{\vec{u} \cdot ...
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675 views

Definition of a Field of Characteristic $n$?

Let $V$ be a vector space over a field of characteristic not equal to $2$. Prove that $\{u, v\}$ is linearly independent with $u, v$ being distinct if and only if $\{u+v, v-v\}$ is linearly ...
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Is there any simple way to write down permutations set-theoretically?

Let $\{J_{n_1}(\lambda_1),...,J_{n_m}(\lambda_m)\}$ and $\{J_{l_1}(\mu_1),...,J_{l_k}(\mu_k)\}$ be finite sequences of Jordan blocks with entries in a field $F$. Let $A\triangleq ...
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46 views

Why in this proof we get $\alpha \geq 0$?

I've solved the following problem: "Let $u,v \in \mathbb{R}^n$ with $u \neq 0$ be such that $|u+v|=|u|+|v|$ (euclidean norm), show that there's $\alpha \in \mathbb{R}$ with $\alpha \geq 0$ such that ...
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49 views

A separation lemma in a real vector space

A lattice $N$ is a free $\mathbb{Z}$-module of finite rank. Let $V$ be the real vector space $N\otimes_\mathbb{Z} \mathbb{R}.$ A cone is a set $\sigma = \{ r_1 v_1 + \ldots + r_k v_k \in V : r_i\geq 0 ...
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96 views

Eigenvalues of $\operatorname{ad}x$

Let $x\in \operatorname{gl}(n,F)$ have $n$ distinct eigenvalues $a_1,\ldots,a_n$ in $F$. Prove that the eigenvalues of $\text{ad }x$ are precisely the $n^2$ scalars $a_i-a_j$ ($1\leq i,j\leq n$), ...
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205 views

Make a matrix invertible

Suppose that $ A $ is $n \times n $ matrix with a 1-dimensional null-space. Show that we can choose vectors $u$ and $v$ so that the linear transformation \begin{equation} B = A + u \otimes v^t ...
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Is this proof that the vectors are colinear correct?

I was solving the following exercise: "Let $x,y \in \mathbb{R}^n$ be nonzero such that if $z$ is orthogonal to $x$ then $z$ is orthogonal to $y$. Prove that $x$ and $y$ are colinear". My idea was: ...
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86 views

Finding the rank of a certain general matrix

If I have an $m \times n$ matrix $A$ and an $n \times m$ matrix $B$ such that $AB=I_m$, how do I go about calculating the rank of $A$ and the rank of $B$? Any clues would be much appreciated!
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69 views

Eigenvector of matrix

Vectors $(1,0,0),(0,1,1),(1,1,1)$ are eigenvectors of matrix $A$. Prove that vector $(1,2,2)$ is eigenvector of matrix $A$. We have: $$A(1,0,0) = \lambda_1 (1,0,0) \\ A(0,1,1) = \lambda_2 (0,1,1) ...
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248 views

Solving commutator equation: $AX-XA=M$

Consider the field square matrices $M_n(\mathbb{C})$ or $M_n(\mathbb{R})$. I wish to solve the equation $AX-XA=M$ for a given $A,M$. Obviously this is just $n^2$ linear equations and thus is trivial ...
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104 views

Identifying the nature of the eigenvalues

I wish somebody could help me in this one. We have to choose one of the $4$ options. Let $a,b,c$ be positive real numbers such that $b^2+c^2<a<1$. Consider the $3 \times 3$ matrix ...
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99 views

Finding an orthogonal basis of the subspace spanned by given vectors

Let W be the subspace spanned by the given vectors. Find a basis for $W^\perp$. $$v_1=(2,1,-2) ;v_2=(4,0,1)$$ Well I did the following to find the basis. $$(x,y,z)*(2,1,-2)=0$$ ...
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Proving Distributivity of Matrix Multiplication

If $A,B,C$ are matrices I am thinking how to show that $$ A(B + C) = AB + AC$$ Is possible to show without sums like $\sum_i a_i, ..., \sum_j b_j$? It seems if I do the proof with many indexes then ...
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93 views

Is this linear algebra proof correct?

I did more linear algebra exercies and I'd appreciate if someone could tell me if my solution is correct. The exercise is: Show that if $T_1,...,T_n$ are injective linear maps (of the right ...
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98 views

A is similar to $A^k$, then each eigenvalue of $A$ is a root of unity

Let $A \in \mathbb{C}(n,n)$ and $k \geq 2$ be an integer such that $$A \sim A^k$$. Show that if $A$ is non-singular then each eigenvalue of $A$ is a root of unity. Attempt: Since $A \sim A^k$, $$PA ...
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79 views

Derivation linear map examples

From Humphreys' Introduction to Lie Algebras and Representation Theory: By an $F$-algebra (not necessarily associative) we simply mean a vector space $U$ over $F$ endowed with a bilinear operation ...
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333 views

Skew-symmetric matrices dot product condition

From the Wikipedia page on skew-symmetric matrices: Denote with $\langle\cdot,\cdot\rangle$ the standard inner product on $\mathbb{R}^n$. The real $n$-by-$n$ matrix $A$ is skew-symmetric if and ...
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151 views

Prove that $f$ is a linear combination of $f_1,f_2,\dots,f_n$.

Let $V$ be a vector space and let $f, f_1,f_2,\dots,f_n$ be linear maps from $V$ to $\mathbb{R}$. Suppose that $f(x)=0$ whenever $f_1(x)=f_2(x)=\cdots=f_n(x)=0$. Prove that $f$ is a linear combination ...
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Proofread matrix multiply inverse of matrix are unique

Prove $\textbf{A}^{-1}$ is unique for $\textbf{A}\textbf{A}^{-1} = \textbf{I}$ Assume $$\textbf{B} \neq \textbf{A}^{-1} \text{ and }\textbf{A}\textbf{B} = \textbf{I}$$ $$\textbf{A}\textbf{A}^{-1} = ...
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57 views

Find the corner of a tetrahedron

The corners A(13, 16, 8) B(-19, -8, 4) C(16, 13, -4) of the tetrahedron ABCD are given and it's know that the centre point M of the circumsphere belongs to the X,Y-plane. The corner D has to be found, ...
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58 views

Is this the correct solution?

Determine the coordinates of the vector $U=(4,5,-3)\;\text{of}\; R^3$ with respect to base ${(1,0,0), (0,1,0), (0,0, 1)}$ $$x(1,0,0) + y (0,1,0) + z (0,0,1) = (4,5, -3)$$ $$(x, 0,0) + (0, y, 0) + ...
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273 views

How to calculate the number of integral points inside an area

How many integral solutions (x, y) exist satisfying the equation |y| + |x| ≤ 4 My approach: I have made the graph after opening the the modulus in the above equation by making four equations. Now ...
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47 views

Is my solution correct?

"Find a basis and its size for $U=\{{(x,y,z)}\in R^{3};3x-y-z=0\}$" My response:$$3x-y-z=0\Longrightarrow y=3x-z$$$$(x,y,z)=(x,3y-z,z)$$Making $x$ and $z$ equals zero, one at a time, we ...
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Can every basis of a vector space be reduced to the standard basis?

Consider the $n$-dimensional vector spaces $V\in\mathbb{R}^n$. Given a non-standard basis $B$ for the vector space $V$, can it be reduced to the standard basis on $\mathbb{R}^n$?
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Triangular matrices and commutators

From Humphreys' Introduction to Lie Algebras and Representation Theory: We conclude this subsection by mentioning several other subalgebras of $gl(n,F)$ which play an important subsidiary role for ...
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Question about linear operators

Let $X$ be an arbitrary vector space over the reals or the complex numbers, and let $L$ be a linear operator on $X$ such that $L^2=I$, where $I$ is the identity operator on $X$. Must it be true that ...
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1answer
63 views

Skew-symmetric matrix/form

What's the relation between a matrix $A$ that represents a skew-symmetric form, and a skew-symmetric matrix $B$? The matrix $A$ is such that for all vectors $v,w$, we have $v^TAw=-w^TAv$. The matrix ...
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296 views

How do I calculate the values of the control points for an uniform cubic B spline surface?

I want to interpolate the following 3 scattered data points: (80.9,58.5,48.0),(35.0,89.6,82.3),(74.7,17.4,85.9) by an uniform cubic B spline surface on the following control lattice, $ \phi $: In ...
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1answer
78 views

One dimensional linear maps are scalar multiplication

I would like to prove that if $T: \mathbb F \to \mathbb F$ is linear where $\mathbb F \in \{\mathbb R, \mathbb C \}$ then $Tv = \lambda v$ for some scalar. It occurred to me that one may prove this ...
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165 views

How it follows that $a=0$

If $a \in \mathbb F$ is a scalar and $v \in V$ is a vector and I prove that if $av = 0$ then $a=0$ or $v=0$ I argue like this: Assume $a \neq 0$. Then multiplyaing both sides with its inverse implies ...
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4k views

Angle between two 3D lines

I know for given 2 vector $\vec{u},\vec{v}$ the angle between them achieved by - $$\cos{\theta} = \frac{\vec{u} \cdot \vec{v}}{\|\mathbf{u}\|\|\mathbf{v}\|}$$ but what if I want to calculate the ...
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1answer
107 views

Find a basis of $V$ containing $v$ and $w$

Find a basis of $V$ containing $v$ and $w$, where $V=\mathbb{R}^4, v=(0,0,1,1), w=(1,1,1,1)$. I am not sure how to begin so a hint would be appreciated. I suspect I must use the following fact: ...
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1answer
61 views

Inverse of a matrix defined on C

Let $ A $ be a real symmetric matrix and form the matrix \begin{equation} R(z)=(zI - A)^{-1} \end{equation} for complex values of z, whenever it is defined. Prove: The elements of R(z) are quotients ...
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383 views

Intuition/Understanding of Inverse and Determinants

This is not homework, but extends from a proof in my book. EDIT We're given an $m \times m$ nonsingular matrix $B$. According to the definition of an inverse, we can calculate each element of a ...