# linear-algebra-eye-catch3

• 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 […] • Every Group of Order 24 Has a Normal Subgroup of Order 4 or 8 Prove that every group of order 24 has a normal subgroup of order 4 or 8. Proof. Let G be a group of order 24. Note that 24=2^3\cdot 3. Let P be a Sylow 2-subgroup of G. Then |P|=8. Consider the action of the group G on […] • Galois Group of the Polynomial x^p-2. Let p \in \Z be a prime number. Then describe the elements of the Galois group of the polynomial x^p-2. Solution. The roots of the polynomial x^p-2 are \[ \sqrt[p]{2}\zeta^k, k=0,1, \dots, p-1$ where $\sqrt[p]{2}$ is a real $p$-th root of $2$ and $\zeta$ […]
• Subspace Spanned by Trigonometric Functions $\sin^2(x)$ and $\cos^2(x)$ Let $C[-2\pi, 2\pi]$ be the vector space of all real-valued continuous functions defined on the interval $[-2\pi, 2\pi]$. Consider the subspace $W=\Span\{\sin^2(x), \cos^2(x)\}$ spanned by functions $\sin^2(x)$ and $\cos^2(x)$. (a) Prove that the set $B=\{\sin^2(x), \cos^2(x)\}$ […]
• Determine Conditions on Scalars so that the Set of Vectors is Linearly Dependent Determine conditions on the scalars $a, b$ so that the following set $S$ of vectors is linearly dependent. \begin{align*} S=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}, \end{align*} where \[\mathbf{v}_1=\begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix}, […]
• If the Images of Vectors are Linearly Independent, then They Are Linearly Independent Let $T: \R^n \to \R^m$ be a linear transformation. Suppose that $S=\{\mathbf{x}_1, \mathbf{x}_2,\dots, \mathbf{x}_k\}$ is a subset of $\R^n$ such that $\{T(\mathbf{x}_1), T(\mathbf{x}_2), \dots, T(\mathbf{x}_k) \}$ is a linearly independent subset of $\R^m$. Prove that the set $S$ […]