query: Proof that if $C_1g(t)+C_2f(t)=0$ then $g(t)=\sin{t}$ and $f(t)=\cos{t}$ is linearly independent.
positive_passages.docid: 18967
positive_passages.text: Linear independence of $\sin(x)$ and $\cos(x)$
negative_passages.docid: 56102
negative_passages.text: Prove that there is no non-zero linear operator on $\mathbb C^{2}$ such that $(\alpha | T\alpha) = 0$
negative_passages.docid: 42495
negative_passages.text: Determine if the following sets of vectors are linearly independent
negative_passages.docid: 30720
negative_passages.text: If the dot product between two vectors is $0$, are the two linearly independent?
negative_passages.docid: 83173
negative_passages.text: Prove linear combinations of logarithms of primes over $\mathbb{Q}$ is independent
negative_passages.docid: 149801
negative_passages.text: prove that $TS = ST$
negative_passages.docid: 146033
negative_passages.text: Intuitive Proof of the Chain Rule in 1 Variable
negative_passages.docid: 25050
negative_passages.text: To prove that $(F,+)$ and $(F-\{0\},\cdot)$ are not isomorphic as groups.
negative_passages.docid: 79457
negative_passages.text: Geometric random variables $X_1:G(p_1)$ $X_2:G(p_2)$ $X_3:G(p_3)$ are independent, prove the following :
negative_passages.docid: 30017
negative_passages.text: Orthogonality and linear independence
negative_passages.docid: 90475
negative_passages.text: Coordinate-free proof of $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$?
negative_passages.docid: 97049
negative_passages.text: If $v_1,...,v_m$ are linearly independent, then the span $v_1+w,...,v_m+w$ has dimension $\ge m-1$
negative_passages.docid: 156890
negative_passages.text: If $f$ and $g$ satisfy the sine/cosine addition formulae, then what is $g'(0)$?
negative_passages.docid: 93121
negative_passages.text: How to prove there exists a linear transformation?
negative_passages.docid: 80279
negative_passages.text: Show that $S + T$ and $aT$ are linear
negative_passages.docid: 29055
negative_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: 135577
negative_passages.text: Prove the existence of $c$ such that $f'(c) = 2c(f(c) - f(0))$