Questions tagged [linear-algebra]
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For questions about 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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Characterization of linear differential operators in $\mathbb{R}^n$
I was wondering exactly how to characterize a linear differential operators in $\mathbb{R}^n$. Some lecture notes on the internet told me that a differential operator of order $m$ in $\mathbb{R}^n$ is ... real-analysis linear-algebra partial-differential-equations- 1,839
Relationship between coordinate systems and linear algebra coordinates
From 2ed of Kunze's Linear Algebra, coordinates are defined as the ordered set of scalars which produce a particular vector via linear combination of basis vectors. In particular, there exist ... linear-algebra- 375
What are the conditions for two vectors to be equal?
I am going through a derivation of a physics theorem (particulars are not important). It involves a sphere centered at the origin. The observation is made that $\hat{n} = \hat{r}$ (unit normal to ... linear-algebra vectors physics- 375
Calculate the signature of $f$
How I can calculate the signature of this inner product? Let $\mathbb{R}$-vector space $\mathcal{M}_{n \times n} ( \mathbb{R}) \rightarrow \mathbb{R} $ defined by: $ f: \space \mathcal{M}_{n \times n} ... linear-algebra bilinear-form bilinear-operator- 21
Normalizer of a subgroup in $\mathbb{Z}_2^{\otimes 2n}$
Does anyone know a fast method/algorithm for calculating the normalizer of an abelian subgroup $G$ of $ \mathbb{Z}_2^{\otimes 2n}$ (equipped with a symplectic inner product)? Or do I need to check ... linear-algebra abstract-algebra- 1
Matrix norm (submultiplicative constant)
Is it true that we can always find some $k>0$ such that $||XY||\leq k||X||\cdot||Y||$ holds? $||\cdot|| $ is any matrix norm. I haven't learn functional analysis yet. real-analysis linear-algebra matrices numerical-linear-algebra matrix-analysis- 11
prove $(\mathbb{U} \oplus \mathbb{W})^{\perp} \subseteq \mathbb{U}^{\perp}$?
Let $\mathbb{V}$ be a n-dimensional vector space with an inner product. And $\mathbb{U}, \mathbb{W}$ be subspace of $\mathbb{V}$, and $\mathbb{U} \cap \mathbb{W}=\{ \vec{0} \}$ How do I prove $(\... linear-algebra- 21
Lower bound on a bilinear quadratic form
Does the inequality $$x^T A y \ge \lambda_{\text{min}}(A) \|x\|\|y\|$$ hold under any conditions? Edit: I am stuck here: Diagonalizing $A=UDU^T$ and letting $d_i,j$ denote the $i,j^{th}$ entry of $D$, ... linear-algebra eigenvalues-eigenvectors upper-lower-bounds- 628
Non existence of a polynomial between two vectorspaces
Let be $V$ the vector space of the sequences in $\mathbb{C}$ and $\varphi: V \rightarrow V,(x_1,x_2,...) \mapsto (x_2,x_3,...)$ Show that there is no polynomial such that $f \in \mathbb{C}[x] \... linear-algebra abstract-algebra- 1,123
How do I express vectors as linear combinations?
Basically, I'm stuck at this question for some time, and it goes like this: A, B, C, and D are consecutive points of a parallelogram. Point E divides the diagonal AC do that |AE|:|EC| = 1 : 3. Point F ... linear-algebra- 41
Definition of Subspace of a Vector space
Following is the definition of subspace of a vector space in Hoffman linear algebra book: Let $V$ be a vector space over the field $F$. A subspace of $V$ is a subset $W$ of $V$ which is itself a ... linear-algebra vector-spaces definition- 1,523
Showing that minimal polynomial of rank $r$ matrix has at most degree $r + 1$.
My attempt: Suppose the degree was $>r + 1$, i.e. $\mathbf{I}, \mathbf{A}, \ldots , \mathbf{A}^{r + 1}$ are linearly independent. This would mean \begin{align*} \forall \alpha _{0}, \ldots , \... linear-algebra solution-verification minimal-polynomials- 63
Matrix of a linear transformation in a different basis
Let $A : \mathbb{R_2}[x] \rightarrow \mathbb{R_2}[x]$be a linear transormation that has in a basis $\{1,x,x^2\}$ a given matrix: $$ \begin{bmatrix} 1 & 1 & 0\\ 0 & 1 & 2\\ 0 & 0 &... linear-algebra- 73
Eigenvectors of submatrices of circulant, symmetric matrices?
Consider a matrix $M \in \mathbb{R}^{n \times n}$ such that $M_{i+1,j+1} = M_{ij}$, (where $+1$ is understood modulo $n$), and $M_{ij} = M_{ji}$. Let $D$ be a diagonal matrix $\in \mathbb{R}^{n \times ... linear-algebra eigenvalues-eigenvectors symmetric-matrices circulant-matrices- 999
Let $U, V$ be vectors. $|U|=5, V$ is unitary & angle between $U$ and $V$ is $\frac{\pi}{6}$, find scalar $λ$ such that $U+λV$ is orthogonal to $U$. [closed]
Let $U, V$ be vectors in some vector space. If $|U|=5$, then $V$ is unitary and the angle between $U$ and $V$ is $\frac{\pi}{6}$, find the scalar $\lambda$ such that $U+\lambda V$ is orthogonal to $U$.... linear-algebra vector-spaces vectors orthogonality- 11
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