# HW_frontpage

by Yu ·

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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 […]