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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Subspaces of vector space and their spans

$\def\sp{\operatorname{sp}}$ Given 2 subspaces of V, and $T\subseteq \sp(K)$ and $K\subseteq \sp(T)$ then $\sp(K)=\sp(T)$? If $ T\subseteq \sp(T)\subseteq \sp(K)$ and $K\subseteq ...
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Proof for pythagoras theorem

Let $f,g$ orthogonals to each other. $${\left\| {f + g} \right\|^2} = \left<f,f\right>+\left<g,f\right>+\left<f,g\right>+\left<g,g\right> = {\left\| f \right\|^2} + {\left\| g ...
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Find a hyperplane not intersecting $S$

I am struggling with the following problem: Let $K$ be an infinite field, $V$ an $n$-dimensional $K$-vector space, $S \subset V$ a finite subset with $0 \notin S$. Prove that there exists a subspace ...
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3answers
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What ring-sum of vector spaces can possibly mean?

I'm given this test assignment, and I can't decipher what it says. Would you kindly help me? Here's the assignment itself: Let $U$ and $W$ be sub-spaces of the linear vector space $V$ s.t. $U ...
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2answers
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Jordan form of a matrix

Let $$A = \left( {\matrix{ 0 & 1 & 0 & 0 \cr 0 & 0 & 2 & 0 \cr 0 & 0 & 0 & 3 \cr 0 & 0 & 0 & 0 \cr } } \right)$$ The ...
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Proving $\mathbf x$ is a $n\times 1$ vector and $\mathbf A$ an $n\times n$ matrix, then $\mathbf x'\mathbf A \mathbf x = \text{tr} (\mathbf {Axx}')$

If $\mathbf x$ is a $n\times 1$ vector and $\mathbf A$ an $n\times n$ matrix, then $\mathbf x'\mathbf A \mathbf x = \text{tr} (\mathbf {Axx}') (\mathbf A'=transpose A) $
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How to find the velocity and accelaration in a 3d space with 6 degrees of freedom?

I have the following rigid body: I assume that the body is a symmetric cylinder.x,y,z are the axes of the reference frame resulting from a transformation involving three orthogonal rotations ...
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164 views

Book comparison, Linear Algebra

so next semester (Spring 2015) I'm taking a Linear Algebra class. I was wondering if anyone who's had this book "Linear Algebra and Its Applications, 4th Edition - by David C. Lay" can give me an ...
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Linear transformation to higher dimensional space.

There is a 7-by-6 matrix $H$ given. Its rank is 6. I'd like to design a 6-by-5 matrix $D$ such that the following holds: $ \left[ \begin{array}{l} l_1(a_1, a_2, a_3, a_4) \\ l_2(a_1, a_2, a_3, a_4) ...
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complex problem in linear algebra

Let $A$ be an $n$ by $n$ matrix. Let $D$ be an $n$ by $n$ diagonal matrix with distinct diagonal entries, and let $u$ be an $n$ by $1$ column vector with all non-zero entries. Let $Aq=\lambda q$ with ...
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Can a low-rank matrix set have nonempty interior?

The answer to this question may be super simple, but it is very not obvious to me. Consider the space $S^n$ of symmetric $n\times n$ matrices. Consider $T\subset S$ the set of rank $n-1$ matrices. ...
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324 views

Proving that $T$:$(x_1,…,x_n) \rightarrow (\frac {x_1+x_2}{2},\frac {x_2+x_3}{2},…,\frac {x_n+x_1}{2})$ leads to nonintegral components

Start with $n$ paiwise different integers $x_1,x_2,...,x_n,(n>2)$ and repeat the following step: $T$:$(x_1,...,x_n) \rightarrow (\frac {x_1+x_2}{2},\frac {x_2+x_3}{2},...,\frac {x_n+x_1}{2})$ ...
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Let $A$, $B$ be two $3\times3$ commuting matrices, where $A$ is nilpotent and $\operatorname{tr}B = 0$. Prove that $ABA = O$

Let $A$ and $B$ be two $3\times3$ commuting matrices, where $A$ is nilpotent and $\operatorname{tr}B = 0$. Prove that $ABA = 0$. Progress I know that $ABA=0 \implies A^2B=0$. Here ...
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What is the geometric interpretation of a vector squared?

I'm working through Introduction to Space Dynamics by William Tyrrell Thomson. I am having to do a lot of research to make it through even small parts, but I am unable to find information to make me ...
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Request for clarification about linear combinations

I need help understanding the basis of this statement in Axler's Linear Algebra Done Right, found on page 86 of the second edition: Because ($\vec{v_{1}}, \ldots, \vec{v_{n}}$) is a basis of $V$, we ...
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56 views

Change of basis matrix - part of a proof

I'm trying to understand a proof from Comprehensive Introduction to Linear Algebra (page 244) I can't really figure out what steps have been taken to get from eq. 1. to eq. 2. It's just ...
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1answer
58 views

Given $n$ linear functionals $f_k(x_1,\dotsc,x_n) = \sum_{j=1}^n (k-j)x_j$, what is the dimension of the subspace they annihilate?

Let $F$ be a subfield of the complex numbers. We define $n$ linear functionals on $F^n$ ($n \geq 2$) by $f_k(x_1, \dotsc, x_n) = \sum_{j=1}^n (k-j) x_j$, $1 \leq k \leq n$. What is the dimension of ...
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To determine Rank of Linear Transformation

Question is to find the rank of $T_1 $and $T_2$ Since the composition is bijective so rank of $T_1T_2 = m$. But how do I get the ranks of$ T_1 $and$ T_2 $from here? Thanks.
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Set of linear equations with coefficients - solution using matrices

I have a set of linear equations: \begin{matrix} ax_{1}& {}+bx_{2}& {}+x_{3}& & =0\\ cx_{1}& {}+dx_{2}& &{}-x_{4} & =0\\ & {}-ex_{2}& ...
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1answer
105 views

Help Understanding Proof from Linear Algebra Done Right

I'm doing a self-study of Axler's Linear Algebra Done Right, and am looking for some help understanding a step in the proof of Proposition 5.21, appearing on page 89 of the second edition. An ...
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152 views

General form for the rotation of a function.

When rotating linear functions, I would approach the task geometrically (find invariant point etc.), yet I tried using a matrix which worked nicely. This was what I did to rotate $y=2x+1$ by ...
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Is $ x^n-y^n$ is a product of coprime factors?

In the expression: $x^n-y^n$, if $n>2$ and $x,y$ are relatively prime, are the factors $x-y$ and $ x^{n-1}+x^{n-2}y+.....$ always coprime? Why? Please exclude the cases where $x-y=\pm 1$ and $\pm ...
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1answer
73 views

How to solve this kind of problem?

I've just found the following problem: $\quad\quad$ $\quad\quad$ $\quad\,$ And I believe that it could be done with something in combinatorics, my feeling is that generating functions would ...
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1answer
121 views

Tensor Product: Vector Spaces

Reference Foundation for: Hilbert Spaces: Tensor Product Problem Given a vector spaces $V$ and $W$. Take its algebraic tensor product: $\tau:V\times W\to V\otimes W$ How to prove that the image ...
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Linear dependency of nilpotent matrices

I would like to prove that four $2\times 2$ nilpotent matrices are always linearly dependent, using the Cayley-Hamilton theorem or the minimal polynomial in some way. I think I have proved the ...
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7answers
136 views

Given matrix P such that $P^{102 } =0 $ , to show that $P^{2} = 0$.

P is given to be a 2×2 matrix such that $P^{102} = 0$. How to show that $P^{2} =0 $?
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If A and A' are approximately the same, are their principal components/SVD very close?

If we have that two matrices $A\approx A'$ within some guaranteed error bound for each term, and $A=U\Sigma V$ is the singular value decomposition for $A$, and $A'=U'\Sigma' V'$ is the SVD for $A'$, ...
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To find the two dimensional subspace of $R^{3}$

I am stuck with this question .Kindly help me to get through this Option A is of 1 dimension so it cannot be answer but all other options are looking fine to me , What i am missing ? THANKS
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2answers
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Using inverse of matrix A as approximate inverse of matrix that is very close to A

Say we have two matrices, $A$ and $A'$ so that $A\approx A'$, and we have the inverse of $A$, $B$, where $AB=I$, and the inverse of $A'$ where $A'B'=I$. If we have some guarantee about how big any ...
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On Rank of matrices

Let $A$ be a $\{0,1\}$ square matrix. Let $J$ be all $1$ matrix. Let $\bar{A}=J-A$. Is $rk(A)\geq rk(\bar{A})-1$ and $rk(\bar{A})\geq rk({A})-1$ always true? We are over $\Bbb R$.
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Null/Col/Row space be a line\plane through the origin?

For a $4\times3$ matrix can the nullspace, the column space and row space all be lines through the origin? For a $2\times4$ matrix can the nullspace, the column space and row space all be planes ...
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Two square matrices with the same minimial polynomial are similar for $n=5$ or $n=6$ [duplicate]

Let $\mathbb{F}$ be a field, $\lambda \in \mathbb{F}$ and $A,B \in M_n(\mathbb{F})$ such that $m_A(x)=m_B(x)=(x-\lambda)^k$ and such that the geometric multiplicity of $\lambda$ in $A$ equals to the ...
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What does adjoint of a linear map?

I have been studying Linear Algebra from Axler, and I came across adjoint of a linear map. I understood the properties and concept of adjoint, basically $\langle Tv,w \rangle = \langle v,T^*w \rangle ...
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51 views

conditions for $A +B$ to be semi-definite.

Suppose $A$ is a positive definite real matrix, and $B$ is symmetric and real matrix with $B_{ii}>0$. Are there conditions on $\sup_{j}|B_{ij}|$ that can guarantee $A+B$ is semi-definite. ...
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Interpretation of a 3 Variable System of Equations

I'm a high school student, and, of course, this week is finals week. For my Algebra 2 semester final, we have been permitted to take the test home and collaborate with others. This final can be viewed ...
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a question regarding wronskian

I was working on following problem: Let $y_1$ and $y_2$ be solutions of $x^2y'' + y' + (\sin x)y = 0$ satisfying $y_1(0) = 0, y_1'(0)=1,y_2(0) = 1, y_2'(0)=0 $. I worked like following: since ...
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1answer
41 views

What can we say about output of Gram–Schmidt process

Given $\{x_1, \dots, x_{n-1}\}$ linearly independent vectors and $x_n \in \operatorname{span}\{x_1, \dots, x_{n-1}\}$ and let $\{\hat{x_1}, \dots, \hat{x_{n-1}}, \hat{{x_n}}\}$ be the output of the ...
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1answer
111 views

Does every isomorphism between $V$ and $V^*$ send some basis to its dual basis?

Suppose that I have a vector space isomorphism $\theta: V \to V^*$ where $V$ is any vector space (probably over $\mathbb{C}$ is required) and $V^*$ is its dual space. Is it always possible to find a ...
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Eigenvalues of hermitian plus skew-hermitian PSD matrix

I was wondering, suppose you have a matrix of the form $A=B+iCC^\dagger$ where $^\dagger$ denotes the hermitian conjugate. $B$ is hermitian and $CC^\dagger$ is obviously hermitian positive ...
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Orthographic projection of point $[0, 0, 0]$

What is the easiest way to calculate orthographic projection of point $[0, 0, 0]$ on a plane given by formula $x - y + z = 1$?
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Every invertible linear transformation can be perturbed a bit without destroying invertbility, Neumann series

Let $T: V \to V$ be any linear transformation on a real or complex vector space $V$. Show that there exists $\epsilon_0 > 0$ $($depending on $T$$)$ so that $I + \epsilon T$ is invertible for any ...
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How can a $k\times (k-m)$ matrix be multiplied by a $k\times m$ matrix?

While reading a book on differential geometry, I came across this line: Since the differential $d\psi_0(x_0):\mathbb R^m\to \mathbb R^k$ is injective, there is a matrix $B\in \mathbb R^{k\times ...
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1answer
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Finding basis of inverse image

Let $\psi $ be a linear transformation such that$$\psi ([x_1,x_2,x_3,x_4])=[x_1+x_3+x_4, -x_2-x_4,x_1+x_2+x_3+2x_4].$$ Find basis of inverse image $\psi^{-1}(W)$ of subspace ...
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Why is this map not surjective at the origin?

$f:\mathbb{R}^{n+1} \rightarrow \mathbb{R}$ $f(x) = |x|^2$ Then the derivative map is $df_{x}(v)=2\sum_{i}{x^iv^i}$ is surjective except at 0. Is it because at 0 df only goes to 0, and doesn't ...
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Find basis of an image

Let $ \psi : \mathbb{R}^4 \rightarrow \mathbb{R}^3$ be a linear transformation described by a formula $$\psi ([x_1,x_2,x_3,x_4])=[x_1+x_3+x_4, -x_2-x_4,x_1+x_2+x_3+2x_4].$$ Find basis of image ...
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1answer
520 views

Physical or geometric meaning of the trace of a matrix

The geometric meaning of the determinant of a matrix as an area or a volume is dealt with in many textbooks. However, I don't know if the trace of a matrix has a geometric meaning too. Is there ...
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Direct sum of two spaces

Let $\alpha_1=[1,1,0,1]$, $\alpha_2=[1,0,1,1], \alpha_3=[1,1,1,1],\alpha_4=[0,1,1,1]$ be a vectors from $\mathbb{R}^4$ let $U=span(\alpha_1, \alpha_2) \ and \ W=span(\alpha_3, \alpha_4)$ Check that ...
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142 views

Determinant: Continuity

Reference Build-up on: Determinant: Definition Problem Given a vector space $V$. Consider an endomorphism $T:V\to V$. Define its determinant $\det:\mathcal{L}(V)\to\mathbb{C}$. Introduce a norm ...
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Where can Gaussian Elimination be used?

I have searched for this and came to know about it that it is traditionally used to solve linear equations, finding determinant, rank of matrix, inverse of matrix. There was a problem on codechef: ...
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a question about general and particular solutions

We have a $3\times 6$ matrix $A$ with rank $3$ (this is all the information we have, no matrix given). Here comes the questions: What is the number of free variables in the solution to the system ...