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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 ... user avatar Steve

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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 ... user avatar Silly Goose

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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 ... user avatar Silly Goose

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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} ... user avatar Luis

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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 ... user avatar BestQuark

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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. user avatar sally wang

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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 $(\... user avatar AlainAllen

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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 $D$, ... user avatar Undertherainbow

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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] \... user avatar Herrpeter

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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 ... user avatar KingKendrick

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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 ... user avatar user264745

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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 , \... user avatar jacob

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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 &... user avatar m259

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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 ... user avatar a_student

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-1votes1answer36views

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 $U$.... user avatar VOZ ESTOICA

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