query: Prove that if $\{u_1, u_2, u_3\}$ is an orthogonal set of nonzero vectors in $\mathbb{R}^n$ and we have $c_1u_1+ c_2u_2+c_3u_3 = 0$, then $c_i=0$.
positive_passages.docid: 152147
positive_passages.text: Why is a set of orthonormal vectors linearly independent?
negative_passages.docid: 118049
negative_passages.text: Show that $\mathbb{R}^{\mathbb{R}} = U_{e} \oplus U_{o}$
negative_passages.docid: 69980
negative_passages.text: Proving that if $A$ is an $n\times n$ orthogonal matrix, then $\det(A) = \pm1$
negative_passages.docid: 52946
negative_passages.text: If $X$ is an orthogonal matrix, why does $X^TX = I$?
negative_passages.docid: 97994
negative_passages.text: Orthogonality Relations Exercise, Brezis' Book Functional Analysis
negative_passages.docid: 84999
negative_passages.text: $\exp(A)$ is orthogonal iff $A$ is skew symmetric
negative_passages.docid: 92291
negative_passages.text: $U_1, U_2, ..., U_k$ subspaces of linear space $X$ with same dimensions, prove $X=U_1\oplus V=U_2\oplus V=...=U_k\oplus V$
negative_passages.docid: 10493
negative_passages.text: Prove any orthogonal 2-by-2 can be written in one of these forms....
negative_passages.docid: 151390
negative_passages.text: Let $q_1,q_2,q_3,q_4$ be orthonormal vectors in $\mathbb{R}^4$,$z_1,z_2,...,z_6$ orthonormal vectors in $\mathbb{R}^6$ and $A=z_1q_1^T + z_2q_2^T$.
negative_passages.docid: 37985
negative_passages.text: Show that $|x|^2 = |y|^2$ with $A$ orthogonal and $Ax = \cos (θ)x − \sin (θ)y$, $Ay = \sin (θ)x + \cos (θ) y$
negative_passages.docid: 69497
negative_passages.text: Show that the set of all $n \times n$ orthogonal matrices, $O(n)$, is a compact subset of $\mbox{GL} (n,\mathbb R)$
negative_passages.docid: 129549
negative_passages.text: Showing that the orthogonal projection in a Hilbert space is compact iff the subspace is finite dimensional
negative_passages.docid: 158852
negative_passages.text: If $\{v_1, v_2, v_3\}$ is an orthonormal basis of $\mathbb{R}^3$ and $v = a_1v_1 + a_2v_2 + a_3v_3$, can we have $|a_1| > 3$?
negative_passages.docid: 158343
negative_passages.text: how to prove an element is non-zero in a tensor-product
negative_passages.docid: 43511
negative_passages.text: Proving products of coefficients of two polynomials are zero when the product of the polynomials is zero
negative_passages.docid: 18477
negative_passages.text: $(W_1+W_2+\cdots+W_n)^a \leq W_1^a +\cdots + W_n^a$ for $n$ integer, $n\geq 2$, $W\gt 0$ and $a$ constant, real, $0\lt a\lt 1$
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