# HW_frontpage

HW_frontpage

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• 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 […]
• The Product of a Subgroup and a Normal Subgroup is a Subgroup Let $G$ be a group. Let $H$ be a subgroup of $G$ and let $N$ be a normal subgroup of $G$. The product of $H$ and $N$ is defined to be the subset $H\cdot N=\{hn\in G\mid h \in H, n\in N\}.$ Prove that the product $H\cdot N$ is a subgroup of […]
• If the Sum of Entries in Each Row of a Matrix is Zero, then the Matrix is Singular Let $A$ be an $n\times n$ matrix. Suppose that the sum of elements in each row of $A$ is zero. Then prove that the matrix $A$ is singular.   Definition. An $n\times n$ matrix $A$ is said to be singular if there exists a nonzero vector $\mathbf{v}$ such that […]
• Image of a Normal Subgroup Under a Surjective Homomorphism is a Normal Subgroup Let $f: H \to G$ be a surjective group homomorphism from a group $H$ to a group $G$. Let $N$ be a normal subgroup of $H$. Show that the image $f(N)$ is normal in $G$.   Proof. To show that $f(N)$ is normal, we show that $gf(N)g^{-1}=f(N)$ for any $g \in […] • Determine Trigonometric Functions with Given Conditions (a) Find a function $g(\theta) = a \cos(\theta) + b \cos(2 \theta) + c \cos(3 \theta)$ such that$g(0) = g(\pi/2) = g(\pi) = 0$, where$a, b, c$are constants. (b) Find real numbers$a, b, c$such that the function \[g(\theta) = a \cos(\theta) + b \cos(2 \theta) + c \cos(3 […] • A Matrix is Invertible If and Only If It is Nonsingular In this problem, we will show that the concept of non-singularity of a matrix is equivalent to the concept of invertibility. That is, we will prove that: A matrix$A$is nonsingular if and only if$A$is invertible. (a) Show that if$A$is invertible, then$A$is […] • 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 […]