# inverse-matrix

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• Cyclic Group if and only if There Exists a Surjective Group Homomorphism From $\Z$ Show that a group $G$ is cyclic if and only if there exists a surjective group homomorphism from the additive group $\Z$ of integers to the group $G$.   Proof. $(\implies)$: If $G$ is cyclic, then there exists a surjective homomorhpism from $\Z$ Suppose that $G$ is […]
• The Sum of Cosine Squared in an Inner Product Space Let $\mathbf{v}$ be a vector in an inner product space $V$ over $\R$. Suppose that $\{\mathbf{u}_1, \dots, \mathbf{u}_n\}$ is an orthonormal basis of $V$. Let $\theta_i$ be the angle between $\mathbf{v}$ and $\mathbf{u}_i$ for $i=1,\dots, n$. Prove that $\cos […] • The Number of Elements Satisfying g^5=e in a Finite Group is Odd Let G be a finite group. Let S be the set of elements g such that g^5=e, where e is the identity element in the group G. Prove that the number of elements in S is odd. Proof. Let g\neq e be an element in the group G such that g^5=e. As […] • Prove Vector Space Properties Using Vector Space Axioms Using the axiom of a vector space, prove the following properties. Let V be a vector space over \R. Let u, v, w\in V. (a) If u+v=u+w, then v=w. (b) If v+u=w+u, then v=w. (c) The zero vector \mathbf{0} is unique. (d) For each v\in V, the additive inverse […] • Are These Linear Transformations? Define two functions T:\R^{2}\to\R^{2} and S:\R^{2}\to\R^{2} by \[ T\left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} 2x+y \\ 0 \end{bmatrix} ,\; S\left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} x+y […] • Idempotent Elements and Zero Divisors in a Ring and in an Integral Domain Prove the following statements. (a) If a\neq 1 is an idempotent element of R, then a is a zero divisor. (b) Suppose that R is an integral domain. Determine all the idempotent elements of R. Definitions (Idempotent, Zero Divisor, Integral […] • A Relation between the Dot Product and the Trace Let \mathbf{v} and \mathbf{w} be two n \times 1 column vectors. Prove that \tr ( \mathbf{v} \mathbf{w}^\trans ) = \mathbf{v}^\trans \mathbf{w}. Solution. Suppose the vectors have components \[\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n […] • Using the Wronskian for Exponential Functions, Determine Whether the Set is Linearly Independent By calculating the Wronskian, determine whether the set of exponential functions \[\{e^x, e^{2x}, e^{3x}\}$ is linearly independent on the interval $[-1, 1]$.   Solution. The Wronskian for the set $\{e^x, e^{2x}, e^{3x}\}$ is given […]