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.

87,196 questions
Filter by
Sorted by
Tagged with
3k views

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: ...
2k views

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? ...
2k views

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 ...
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 ...
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 ...
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 $a_i$. $f_1 = u g_1$ ...
62 views

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 $\begin{bmatrix} \cos y\\ \sin y \end{bmatrix}$. ...
811 views

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 $c^{(3)}/c_0$. ...
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 ...
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?
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 ...
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 ...
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 ...
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 ...
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 $\text{dim} \ c/c_0$? ...
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 ...
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}(...
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
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 ...
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 ...
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 ...
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 ...