query: Given linearly dependent set $U=\{x, Bx, B^2x,..,B^nx\}$ where $B$ is $n \times n$ matrix prove that matrix $k_0I+k_1B+..+k_nB^n$ is not invertible.
positive_passages.docid: 29055
positive_passages.text: How to prove that $c_0I+c_1A+..+C_nA^n$ is singular if $(v,Av,A^2v,...,A^nv)$ is linearly dependent, $v \in \mathbb{F^n}$ and $A_{n \times n}$?
negative_passages.docid: 88442
negative_passages.text: Prove that there's no ordered basis $E$ in which $T{x\choose y}={0\choose y}$ can be represented as $1\ 2\choose 2\ 4$
negative_passages.docid: 130765
negative_passages.text: Linear dependence of $\left\{x^{n}\,\colon\, n\in\mathbb{N}\right\}$
negative_passages.docid: 46698
negative_passages.text: Showing a set is closed under inverses
negative_passages.docid: 93923
negative_passages.text: No surjective bounded linear map from $\ell^2(\mathbf{N})$ to $\ell^1(\mathbf{N})$
negative_passages.docid: 30017
negative_passages.text: Orthogonality and linear independence
negative_passages.docid: 64130
negative_passages.text: Prove that $\mathbb{Q}^{\times}$ not isomorphic to $\mathbb{Z}^{n}$
negative_passages.docid: 119032
negative_passages.text: Is ${\bf x^TAx} \ge { a \bf x^Tx}$ for some $a>0$ not dependent on $\bf x$ if $\bf A$ is positive definite matrix?
negative_passages.docid: 39334
negative_passages.text: Non-degenerate bilinear forms and invertible matrices
negative_passages.docid: 138343
negative_passages.text: How to prove that the set $\{\sin(x),\sin(2x),...,\sin(mx)\}$ is linearly independent?
negative_passages.docid: 75066
negative_passages.text: If the product of two square matrices is invertible, then both matrices are invertible
negative_passages.docid: 115582
negative_passages.text: Prove that if a product$ AB$ of $n\times n$ matrices is invertible, so are the factors $A$ and $B$.
negative_passages.docid: 6291
negative_passages.text: Proving something is a linearly independent subset
negative_passages.docid: 142229
negative_passages.text: Prove that if $V \cap \text{Span}\{u_1, u_2,...,u_k\} = \{ 0\} $, then $Au_1, Au_2,...,Au_k$ are linearly independent
negative_passages.docid: 66500
negative_passages.text: Proving that orthogonal vectors are linearly independent
negative_passages.docid: 107248
negative_passages.text: Prove that the set $ \{\sin(x),\cos(x),\sin(2x),\cos(2x)\}$ is linearly independent.
negative_passages.docid: 96354
negative_passages.text: Linear independence of $\sin(x)$, $\sin(2x)$, $\sin(3x)$ in Map($\mathbb{R},\mathbb{R}$)