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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Eigenvalues of discretized linear integral operator

Suppose I have the following kernel operator: $Af(x) = \int_{-1}^1 K(x-y)f(y)dy$ which is also positive and compact. Hence, it has a countable set of positive eigenvalues. Suppose those eigenvalues ...
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475 views

Why is it true that $\mathrm{adj}(A)A = \det(A) \cdot I$?

This is a statement in linear algebra that I can't seem to understand the proof behind. For a square matrix $A$, why is: $$\mathrm{adj}(A)A = \det(A) \cdot I$$ Any explanation would be greatly ...
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2k views

How does one go about proving that something is a vector space?

So I have this pretty theoretical problem for homework that says I need to show that this set of matrices is a vector space. It says that we have the set $M_{m,n}(\mathbb R)$, $m\times n$ matrices ...
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1answer
130 views

A family of $n$ non-zero vectors of an $(n-1)$-dimensional vector space must be linearly dependent

I was bored earlier and began to think of the pigeonhole principle, and it came to me that it could be used to show that a family of $n$ non-zero vectors of an $(n-1)$-dimensional vector space must be ...
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1answer
192 views

Change of basis when given linear transformation in another basis

This is a follow up to my previous question. I am studying conjugation/similarity transformations in my introductory linear algebra class. Here is an example: $$ \big[\,T\,\big]_{\mathcal{B}} = ...
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319 views

Multiple Choice question about an $n \times n$ matrix $A$ with real or complex entries, and such that $A^3=0$

Let $A$ be an $n \times n$ matrix with real or complex entries and such that $A^3=0.$ Which of the following options holds? 1. $(I+A)^3=0$. 2. $I+A$ is invertible. 3. $I+A$ is not invertible. ...
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Linear Dependence problem

Lets suppose we have a set of vectors $\{(1 ,0, 0, 0 ) , (0, 1, 0, 0 ), (2, 0, 0, 0 )\}$. By definition this set is linearly dependent because we can find constants $c_1,c_2, c_3$ ( such that all ...
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189 views

Basis of the subspace of $\mathbb R^4$

Find a basis of the subspace of R4 consisiting of all vectors of the form: $$\begin{bmatrix}x_1\\ 6 x_1 + x_2\\ 4 x_1 + 5 x_2\\ 8 x_1 - 9 x_2\end{bmatrix}$$ Now, I really have no clue how to set ...
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1answer
58 views

How to estimate (if/any) displacement/rotations between 2d line segments taken from 2 data sets

I am having set of pair of line segments (2D). Though each pair should be coincided on top of each other they are not so. I derive these two line sets using image based (e.g. CD) and manual method ...
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1answer
41 views

Finding a matrix fixed by this permutation

I have an $n\times n$ matrix $M=1-\sigma(g)$ where $\sigma:S_n\to\mathbb{C}^n$ is a representation (reducible) of the symmetric group. I want to find another matrix $A(g)$ such that $MA(g)=0.$ ...
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Finding a linear transformation $L$ such that $ L(\vec{u}_{i}) = {v}_{i}$ for $ 1 \leq i \leq 3$.

I've been tearing my hair out for hours over this seemingly straightforward linear question. I know it's bad form to post homework without some partial amount of work done, but my work is garbage. I ...
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94 views

Proofs Invertible & Diagonal matrix

Given: $P$ is an invertible matrix. $D$ is a diagonal matrix. $A$ is an $n\times n$ matrix. AND $A = PDP^{-1}$ Prove that the determinant of A equals the product of the diagonal entries of $D$.
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619 views

Proofs: Subspaces, vectors.

Studying for an exam, came across this question: Let $V$ be a subspace of $\mathbb{R}^n$ and $u \in \mathbb{R}^n$ but $u \notin V$. $W = \{v + cu: v$ is in $V$ and $c$ is a scalar$\}$ A) Prove ...
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54 views

Proving different coordinates of same matrix are similar

Let $\mathcal{B}$ and $\mathcal{C}$ both be bases for $\mathbb{R}^n$. If $L:\mathbb{R}^n\to \mathbb{R}^n$ is any linear operator, then prove that $[L]_{\mathcal{B}}$ and $[L]_{\mathcal{C}}$ are ...
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684 views

Determining if a Matrix is Diagonalizable without computing Eigenvalues

Is there any simple way to determine if a matrix is diagonalizable without having to compute eigenvalues? I'm motivated by the idea that for $\mathbb{R}^n$, to determine if a matrix is ...
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162 views

Confusion related to derivative of a quadratic equation

This might be a simple question but how come F(x) = x'Ax F'(x) = (A + A')x I didn't get it
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157 views

What is the name for a non-square permutation matrix?

Consider a matrix that selects and permutes some but not all of the entries of a vector. That is a binary $n\times m$ matrix, where $n<m$, with a single one per row, for example ...
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Clockwise Linear Transformation

I have a problem understanding this. Question: Find the matrix $A$ of the linear transformation from $\Bbb R^2 \to \Bbb R^2$ that rotates any vector through an angle of $150^\circ$ in the clockwise ...
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1answer
83 views

how to proove function satisfy Lipschitz condition

Let $v_1,\ldots, v_n\in \mathbb{F}_{2}^r$ be $n$ column vectors chosen uniformly and independently at random from ${\mathbb{F}_2}^r$. Let $f$ be a function defined as follows:$ [f(v_1,\ldots, v_n)$= ...
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114 views

How do we know the rank is 1?

Here's matrix A: $$ \begin{pmatrix} 2 & 1 \\ -4 & -2 \\ -2 & -1 \end{pmatrix} $$ Apparently, we can determine that matrix A is of rank 1 by noting that. $$ \det \begin{pmatrix} 2 & ...
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1k views

Acute angle between two planes

The acute angle between two planes is called the dihedral angle. Plane $x - 3y + 2z = 0$ and plane $3x-2y-z+3 = 0$ intersect in a line and form a dihedral angle $\theta$ . Find a third plane (in ...
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109 views

Beer brewing formula question - solving for the wrong thing

I am brewing beer at home and I am having trouble with a formula. I know the formula for how to solve for how much water is left in the kettle after boiling the wort; however, this formula isn't much ...
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653 views

Describe all matrices similar to a certain matrix.

Math people: I assigned this problem as homework to my students (from Strang's "Linear Algebra and its Applications", 4th edition): Describe in words all matrices that are similar to ...
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479 views

Volume of the intersection of ellipsoids

How do I compute the volume of the intersection of two $n$-dimensional ellipsoids? Given an $n$-vector $c$ and a symmetric positive-definite $n\times n$ matrix $A$, define the ellipsoid ...
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How does linear algebra help with computer science

I'm a Computer Science student. I've just completed a linear algebra course. I got 75 points out of 100 points on the final exam. I know linear algebra well. As a programmer, I'm having a difficult ...
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2answers
77 views

Tensor products of exterior powers

For my advanced linear algebra course, I have to prove that $(\wedge^{k}V \otimes V \otimes V) \cap (V \otimes \wedge^{k}V \otimes V) = \wedge^{k+1} V \otimes V$, regarded as subspaces of $V^{\otimes ...
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1answer
682 views

tensor product and wedge product for direct sum decomposition

If we have a real vector space $V=W_1\oplus W_2$, is it true that $W_1 \otimes W_2 = W_1 \wedge W_2 $? My guess is that this is true. The definition of the $k$-exterior power is the quotient of ...
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3answers
444 views

Find a $2$ by $3$ system $Ax=b$ whose general solution is…

I came across a problem in my Linear Algebra book that says: Find a $2$ by $3$ system $Ax=b$ whose general solution is $x=\begin{pmatrix} 1\\ 1\\ 0 \end{pmatrix}+w\begin{pmatrix} 1\\ 2\\ 1 ...
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134 views

Quick question regarding matrices of bilinear forms (finding an orthogonal basis)

Ok, I've got the form $\langle A,B \rangle = tr(AB)$ and the vector space of real $2 \times 2$ matrices. The question wants me to determine the matrix of the form with respect to the standard basis of ...
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71 views

Let S = $\{(x, y, z) \;\text{which spans}\;\Bbb R^3\;|\; 2x = 3z \;,\;\; y = -z\}$

I can't find any similar examples in my text book on this. I'm assuming that from this I can either have $x = 3/2z$ and $y = -z$ then have the set $\{3/2z, -z, z\}$ but I need to prove that this is a ...
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402 views

Matrices for change of basis linear transformations

Find the $\mathcal{B}$ matrix of the linear transformation $T: \mathbb{R}^3 \to \mathbb{R}^3$ defined by reflection through the $x_1 x_2$ plane for the basis $\mathcal{B} = \{b_1, b_2, b_3\}$. I ...
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1answer
456 views

dim$(V)$ = $n$, dim$(W)$ = $m$ $\implies$ dim($L(V,W)$) = $nm$

I am reading Hoffman & Kunze's chapter on linear transformations, with a view towards understanding dual spaces. (I primarily want to read Calculus on Manifolds; in the first chapter of that book, ...
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2answers
2k views

equation of a plane that is perpendicular to a line segment.

Find the general equation of the plane which is perpendicular to the line segment between the points $A(1, 2, 9)$ and $B(3, 4, 12)$ that separates the line segment into $2$ equal parts. This is called ...
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1answer
273 views

Is M (a non-symmetric matrix) positive definite if the product NM is positive definite where N is a diagonal positive definite matrix.

If the product of two matrices, N (a diagonal positive definite matrix) and M (a non-symmetric matrix), is positive definite i.e. $x^TNMx>0$, then is the matrix M positive definite i.e. is ...
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100 views

A subspace of a vector space

A subspace of a vector space $V$ is a subset $H$ of $V$ that has three properties: a) The zero vector of $V$ is in $H$. b) $H$ is closed under vector addition. That is for each $u$ and $v$ in $H$, ...
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1answer
237 views

Is my statistician friend right/wrong on metric spaces and norms?

I was talking to a statistician friend of mine who said that instead of minimizing this function $\sum_{i,j}W_{ij}d_{ij}^2(X)$ over $X$ it would be better to solve an analogous minimization problem ...
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4answers
2k views

Distance between planes

Find the distance between the planes $$x + 2y +2z = 4$$ $$z= -\frac12 (x-1)-(y-2)+3$$ First of all how do you check if they are parallel? The integers in plane two are leading me astray? How do I ...
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1answer
304 views

Hugely ugly messy determinant - any trick to find it?

Find the determinant of $$\begin{bmatrix}1 & a & a^2 & a^{3}\\ 1 & b & b^{2} & b^{3}\\ 1 & c & c^{2} & c^{3}\\ 1 & d & d^{2} & d^{3} \end{bmatrix}$$ ...
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1answer
80 views

Linear Algebra Row Space solution

Let's suppose we have a matrix A mxn that implements a linear system Ax=b. If b is inside the Column Space of A, then thel linear system has AT LEAST one solution x in R^n. But every x in R^n can be ...
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4answers
98 views

Characteristic polynomial of an involution

I'm attempting to prove that the characteristic polynomial of an involution in $\mathbb{R}^n$, i.e. a linear transformation that satisfies $f^2=I$ will always factor into linear factors. We know that ...
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Help me create formula for such sequence

:) I'm web-developer, and currently I'm looking for formula to automate "discount" percent discovery. Our old formula is min(10, 5 + ordersDone), which generally ...
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Building a norm.

I've been told to build a norm that is NOT a matrix norm. I need to show that the built norm is indeed a norm on the space of $n*n$ matrices but that is not a matrix norm induced by some vector norm. ...
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1answer
140 views

Is this matrix function convex or non-convex?

Given, $g(Z)=Tr(Z^Tf(Z)Z)$ , where $f(Z)=h(Z)-ZZ^T$ is a p.s.d matrix formed using entries in $Z$, where again $h(Z)$ is a diagonal matrix with its $i$'th diagonal entry being ...
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1answer
120 views

What's the benefit of solving a diagonally dominant matrix compared to an ordinary one?

I have two linear systems of equations. One is strictly diagonal dominant and other is just an ordinary matrix. Both of them could have a very large scale. I'm wondering the benefit of solving a ...
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1answer
57 views

Trace identity deduction - linear algebra.

In a lemma building up to the proof of Cartan's criterion for solubility they deduce something that I don't follow. This is likely due to some deficit in my knowledge of linear algebra. I'll outline ...
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2answers
542 views

Is a bra the adjoint of a ket?

The instructor in my quantum computation course sometimes uses the equivalence $$(\left|a\right>)^\dagger\equiv\left<a\right|$$ I understand that this is true for the typical matrix ...
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4answers
555 views

Eigenvalues of a matrix with only one non-zero row and non-zero column.

Here is the full question. Only the last row and the last column can contain non-zero entries. The matrix entries can take values only from $\{0,1\}$. It is a kind of binary matrix. I am ...
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76 views

Polynomial with bounded coefficients

Given $1<\beta<2$, I need to construct a polynomial (L can be chosen freely )$f(x)=x^{L}+a_{L-1}x^{L-1}+a_{L-2}x^{L-2}+\cdots+a_{1}x^{1}+a_0$ such that it satisfies $$\sum_{k=0}^{L-1}|a_k|=1$$ ...
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1answer
84 views

Disable one angle of rotation

I'd like to disable one angle of rotation of an object rotating in 3D space. Imagine a camera rotating around and displaying objects as they are in space. I'd like this object to be fixed on the ...
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49 views

Representing an imperfect ellipse in 2 linear variables

I have several shapes which are roughly elliptical. I know the major and minor axes and the true circumference, so I store them like this: $$a={\text{axis}}_{\text{major}}\\ ...