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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How many presentable boolean functions with n attributes are linear separable?

The aim is to find a formula for the question. For $n=2$ i get $2^{2^n}=16$ possible functions. This is the solution for a boolean function with 2 attributes: ...
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3answers
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Decomposition of vector with respect to direct sum of vector subspaces

$U$ and $W$ are two subspaces of vector space $V$. If $U \oplus W = V$, then $\forall v \in V$, there exist two unique vectors $u \in U$ and $w \in W$ such that $v = u + w$. Is the reverse true? ...
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1answer
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Cost of Solving Linear System

As most of us are aware the cost for solving a linear system ("exactly") with Gauss Elimination and other similar methods with a few right hand side and where the matrix has no structure is $\mathcal{...
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2answers
397 views

Calculate the real eigenvalues and affiliated eigenspace of a square matrix whose elements are all 1

Question: Let $A \in M(n,n,\mathbb{R}), a_{ij} = 1$ for all $i,j$. Calculate the real eigenvalues and the affiliated eigenspace of $A$. So first of all what I would be trying to calculate are values ...
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3answers
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Show that if A has a right-inverse, then $Ax = b$ has at least one solution for every choice of b in $R^n$

Show that if A has a right-inverse, then $Ax = b$ has at least one solution for every choice of b in $R^n$.
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3answers
255 views

Why is it that if A is an n by m matrix, and both BA and AB are indentity matrices, then A is square and can't be rectangular?

Some background: I was in class today, and the professor was proving that given a matrix $A$ that is $n$ by $m$ and a $B$ such that $AB=I$ where $B$ is obviously $m$ by $n$ and $I$ is $n$ by $n$, if ...
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1answer
139 views

Conditions on matrix expression

What conditions are needed on the matrices A and B so that the following is true: $A^{T}BA=B$. Clearly, if B is invertible, the determinant of A must be 1 or -1. What other conditions are needed?
3
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2answers
129 views

Suggest a tricky procedure

If $$A = \frac{1}{2} \times \begin{pmatrix} -1& -\sqrt3 & 0 \\ -\sqrt3& 1 &0 \\ 0 & 0 & 0 \end{pmatrix} \text{ and } E = \frac{1}{2} \times \begin{pmatrix} 1&...
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2answers
848 views

Matrices with real entries such that $(I -(AB-BA))^{n}=0$

I was just trying out some problems, when i couldn't solve this question: Does there exist $n \times n$ matrices $A$ and $B$ with real entries such that $$ \Bigl(I - (AB-BA)\Bigr)^{n}=0?$$ I really ...
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2answers
201 views

How many ways to choose $l$ vectors in $n$-dimensional space such that every $k$-subset is independent

Working in $F_q^n$. How many different ways do we have to choose $l$ vectors such that every subset of size $k$ of them is linearly independent. (Assume n is large) My Progress: For the first k ...
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1answer
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transpose of positive matrix is positive

how to prove it? I am talking about matrixes which satisfy: $$( Ax , x ) > 0\quad \text{ for any}\quad \;x \neq 0.$$ How to prove that $A^T\;$ is also positive? $$x^T A x = ( x^T A x )^T$$ ...
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4answers
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Sylvester's determinant identity

Sylvester's determinant identity states that if $A$ and $B$ are matrices of sizes $m\times n$ and $n\times m$, then $$ \det(I_m+AB) = \det(I_n+BA)$$ where $I_m$ and $I_n$ denote the $m \times m$ ...
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10answers
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Inverse of the sum of matrices

I have two square matrices - $A$ and $B$. $A^{-1}$ is known and I want to calculate $(A+B)^{-1}$. Are there theorems that help with calculating the inverse of the sum of matrices? In general case $B^{-...
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2answers
571 views

Which is the best method to find extremal eigenvalues of a symmetric matrix?

My question is as to which is the best method to find extremal eigenvalues of a real symmetric matrix? Currently I am using Lanczos Iterations followed by Bisection Method. Does anyone have a better ...
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1answer
50 views

Partition $Z_2^n \setminus \{0\}$ into disjoint sets. Does one contain a maximal subspace?

Let $n \geq 1$. $V=Z_2^n \setminus \{0\}$. $A \cup B = V$. $A \cap B = \Phi$. Is it true that either A or B contains a sub-space of dimension $n-1$ (without the zero element)? (I reduced a homework ...
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2answers
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Defining the determinant of linear transformations as multilinear alternating form

Here is what our professor showed us in our linear algebra class to introduce the idea of determinants: Suppose we have an $n$-dimensional vector space $V$. Then we can create a function from $V^n$ ...
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2answers
547 views

Transform a symmetric matrix into a symmetric matrix of different size

Let $ABC = D$ where $B$ and $D$ are symmetrical matrices. However their [rows x columns] values are not same. For example, $B$ is 2x2 and $D$ is 3x3 a matrix. Clearly, in this case, $A$ has to be a ...
3
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1answer
960 views

Jacobian of a Matrix

Usually Jacobian matrix encapsulate partial derivatives of a vector f w.r.t. another vector v. More generally how to define/get the Jacobian matrix of a matrix F w.r.t another matrix V? Is it just ...
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4answers
1k views

Vector space over $\mathbb{Q}$ or $\mathbb{Z}$?

I am looking at the following: Show that a torsion-free divisible group $G$ is a vector space over $\mathbb{Q}$. I have no problem verifying the axioms of vector spaces after noting that the term ...
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4answers
14k views

Why are all homogenous systems consistent?

A linear system of form $A\vec{x}=\vec{0}$ is called homogeneous. Why are all homogenous systems consistent?
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1answer
130 views

Problem involving cross products

I have a problem in vector algebra. In this Wolfram-page the last two formulas (9) and (10) are: $$ \frac{ | (x_2 - x_1) \times (x_1 - x_0) | }{|x_2 - x_1 |} = \frac{ | (x_0 - x_1) \times (x_0 - ...
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1answer
326 views

Linear operator categories

Let's consider linear operators on the set of complex-valued functions to the same set. I wonder to which categories such operators can be classified. All linear operators I encountered so far fall ...
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2answers
2k views

M-Matrix characterization for the transpose

A common characterization of M-matrices are non-singular square matrices with non-positive off-diagonal entries, positive diagonal entries, non-negative row sums, and at least one positive ...
4
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1answer
574 views

Maximal number of vectors with (pairwise) negative scalar product

Consider $\mathbb{R}^n$ equipped with the standard scalar product. Let $f(n)$ denote the maximal cardinality of a set of vectors in $\mathbb{R}^n$ with a pairwise negative scalar product. What is $f(...
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3answers
651 views

Scale Operator $Uf(x)=f(kx)$

I am looking for an operator $U$, that can do this to a function: $$Uf(x)=f(2x).$$ In particular I am happy if there is an $U$ for the general case: $Uf(x)=f(kx)$. Does such an operator exist for ...
3
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2answers
456 views

A distance preserving operator that's not linear?

Let $Q: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be an operator that preserves all distances. Is this condition alone enough for us to say that $Q$ must be a linear operator? If not, what are some ...
11
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1answer
6k views

In-place inversion of large matrices

In Solving very large matrices in "pieces" there is a way shown to solve matrix inversion in pieces. Is it possible to apply the method in-place? I am refering to the answer in the ...
2
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1answer
127 views

Equations over $\mathbb Z/p^e \mathbb Z$

Let $\{f_i\}$ be a system of linear equations in $X_1,...,X_n$ with coefficients in $R = \mathbb Z/p^e \mathbb Z$ (i.e. modulo a primepower). Assume there is a unique solution $a_i$. $f_1 = u g_1$ ...
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1answer
62 views

How to show that $\hat{e}_m(k) = e^{−2πimk/N}$ for all $k$? How to show that $\hat{F}_m = e_m$?

Let $\{e_0, e_1, \dots, e_{N-1} \}$ be the Euclidean basis for $l^2(Z_N)$, and let $\{F_0, F_1, \dots, F_{N−1} \}$ be the Fourier basis( where $F_m(n)= \frac{1}{N} e^{2 \pi i m n/N}$ and $\hat{z}(m) = ...
2
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2answers
1k views

Solving 12 equations with 13 variables

$16a + 23b + 12c + 34d -2e -37f + 109g -141h + 139i + 149j + 29k + 12l + 131m = 74608$ $13a + 31b -5c + 17d + 29e -67f -16g -101h -7i -201j + 32k + 17l + 171m = -4194$ $81a -12b -5c + 17d -9e + 17f ...
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2answers
3k views

How do I graph a vector [cos x, sin x]?

how do I do the following: Consider the matrix $\begin{bmatrix} \cos x & -\sin x\\ \sin x & \cos x \end{bmatrix}$ and the vector $\begin{bmatrix} \cos y\\ \sin y \end{bmatrix}$. ...
2
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3answers
811 views

Orthogonal and symmetric Matrices

What can one say about the set of all $n$-dimensional square matrices $A \in \text{GL}_n(\mathbb{C})$ that have an inverse with entries out of $\mathbb{C}$ with the properties: unitary $:\...
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1answer
478 views

transformation matrices and complex functions as projections

This question is about the connection between linear algebra and complex analysis. Coming from a two real dimensional domain a transformation matrix geometrically transforms a set of points (e.g. a ...
5
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2answers
9k views

Orthonormal Eigenbasis

I am a little apprehensive to ask this question because I have a feeling it's a "duh" question but I guess that's the beauty of sites like this (anonymity): I need to find an orthonormal eigenbasis ...
3
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1answer
2k views

Zero State, Stable Equilibrium, Dynamic System

Could someone please help? The question reads: For which real numbers $k$ is the zero state a stable equilibrium of the dynamic system $x_{t+1} = Ax_t$? $A = \begin{bmatrix} 0.1 &k \\ 0.3 & ...
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2answers
117 views

Dimension and sequences

Consider the linear space $c^{(3)}$ of all sequences $x = (x_n)_{n=1}^{\infty}$ such that $\{x_{3k+q} \}_{k=0}^{\infty}$ converges for $q = 0,1,2$. Find the dimension and a basis for $c^{(3)}/c_0$. ...
3
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1answer
203 views

Show the matrix product $U = (I+T+iS)(I-T-iS)^{-1}$ is unitary

$S$ is real symmetric. $T$ is real skew-symmetric. I have shown that $T\pm iS$ is skew-Hermitian. I am further asked to show that $U = (I+T+iS)(I-T-iS)^{-1}$ is unitary. Denoting by $^\dagger$ the ...
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2answers
126 views

Can any statement be made on the structure of a diagonal matrix after an unitary transformation?

Let $U$ be unitary and $D$ be diagonal. Can anything be stated about the structure of $$U^\dagger D U$$ then? Also, would this change for an infinite matrix, i.e. a discrete linear operator?
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3answers
885 views

Given a vector and a basis, how can I find the coordinate vector?

I am given a vector space $V$, a basis $e_1,\dots,e_n$ and a vector $v\in V$ and I am asked to find $\lambda_1,\dots\lambda_n$ from the underlying field such that $v=\sum \lambda_i e_i$. How can I do ...
4
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3answers
17k views

Eigenvalues and “Eigenvectors” of Linear Transformations

I am so lost on this question I am not sure even where to start. I am not looking for an answer but more of a turn in the right direction. Find all the eigenvalues and eigenfunctions of the ...
9
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3answers
5k views

Linear Algebra, cube & dimensions > 3

I have found interesting problem in Gilbert's Strang book, ,,Introduction to Linear Algebra'' (3rd edition): How many corners does a cube have in 4 dimensions? How many faces? How many edges? A ...
11
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3answers
4k views

Proof of: $AB=0 \Rightarrow Rank(A)+Rank(B) \leq n$

As the title says, am searching for a proof of If $A,B \in \mathbb{R}^{n\times n}$ and $AB=0$ then $\mathrm{rank}(A)+\mathrm{rank}(B) \leq n$ I am doing this as preparation for an upcoming ...
2
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1answer
143 views

Codimension and sequences

Let $c$ be the set of all convergent sequences and $c_0$ the set of sequences tending to zero. How would one find the codimension of $c_0$? In particular, how would one find $\text{dim} \ c/c_0$? ...
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4answers
4k views

How to check if a subset is a generator of a vector space

I have a very noob questions about generators: what algorithm do I have to follow so I can prove that a finite subset is a generator? Here is the background story (I'll tell it all because I suck at ...
5
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2answers
493 views

Does $M_n^{-1}$ converge for a series of growing matrices $M_n$?

$M_n$ is a $n\times n$ matrix with $M_{n+1}=\begin{pmatrix}M_n & a_n \\ b_n^T & c_n\end{pmatrix}$ and $a_n, b_n, c_n \to 0$ for $n\to \infty$. Is this sufficient to state $$ \lim_{n\to\infty}(...
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1answer
185 views

Embed an $n\times n$ matrix into $R^{n^2}$

How to embed a matrix, for example, a $3x3$ singular matrix in to $R^9$? How to compute the induced metric? Is it just the Frobenius norm of the matrix? Many Thanks. sam
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4answers
3k views

Path for learning linear algebra and multivariable calculus

I'll be finishing Calculus by Spivak somewhat soon, and want to continue into linear algebra and multivariable calculus afterwards. My current plan for learning the two subjects is just to read and ...
5
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0answers
224 views

linear system solution, iterative vs direct

Dear all, I have systems like $(A - \lambda B) X = F$ where lambda is being updated inside a loop. I also have a limited number of eigenvectors of the matrix pair (A, B), say 40 eigenpair from a ...
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1answer
234 views

What does $b_i\mid b_{i+1}$ mean in this context?

In the computational topology literature, the reduction algorithm for computing the Smith normal form of a boundary matrix uses the notation $b_j > 1 \: \text{ and }\: b_j\mid b_{j+1}$ in the ...
3
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1answer
497 views

Moore-Penrose inverse invariance

I have an $n\times p$ matrix $Z$ with $p\gt n$. I have a diagonal matrix $A$ with positive entries. Is there is a known way to determine the MP inverse of $A Z^T Z A$, if I know $A$ and the MP ...