# Galois-theory-eye-catch

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• Condition that Two Matrices are Row Equivalent We say that two $m\times n$ matrices are row equivalent if one can be obtained from the other by a sequence of elementary row operations. Let $A$ and $I$ be $2\times 2$ matrices defined as follows. $A=\begin{bmatrix} 1 & b\\ c& d \end{bmatrix}, \qquad […] • The Polynomial x^p-2 is Irreducible Over the Cyclotomic Field of p-th Root of Unity Prove that the polynomial x^p-2 for a prime number p is irreducible over the field \Q(\zeta_p), where \zeta_p is a primitive pth root of unity. Hint. Consider the field extension \Q(\sqrt[p]{2}, \zeta), where \zeta is a primitive p-th root of […] • Prove a Given Subset is a Subspace and Find a Basis and Dimension Let \[A=\begin{bmatrix} 4 & 1\\ 3& 2 \end{bmatrix}$ and consider the following subset $V$ of the 2-dimensional vector space $\R^2$. $V=\{\mathbf{x}\in \R^2 \mid A\mathbf{x}=5\mathbf{x}\}.$ (a) Prove that the subset $V$ is a subspace of $\R^2$. (b) Find a basis for […]
• Basis of Span in Vector Space of Polynomials of Degree 2 or Less Let $P_2$ be the vector space of all polynomials of degree $2$ or less with real coefficients. Let $S=\{1+x+2x^2, \quad x+2x^2, \quad -1, \quad x^2\}$ be the set of four vectors in $P_2$. Then find a basis of the subspace $\Span(S)$ among the vectors in $S$. (Linear […]
• Determine the Dimension of a Mysterious Vector Space From Coordinate Vectors Let $V$ be a vector space and $B$ be a basis for $V$. Let $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5$ be vectors in $V$. Suppose that $A$ is the matrix whose columns are the coordinate vectors of $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, […] • Eigenvalues of a Hermitian Matrix are Real Numbers Show that eigenvalues of a Hermitian matrix$A$are real numbers. (The Ohio State University Linear Algebra Exam Problem) We give two proofs. These two proofs are essentially the same. The second proof is a bit simpler and concise compared to the first one. […] • Group of Invertible Matrices Over a Finite Field and its Stabilizer Let$\F_p$be the finite field of$p$elements, where$p$is a prime number. Let$G_n=\GL_n(\F_p)$be the group of$n\times n$invertible matrices with entries in the field$\F_p$. As usual in linear algebra, we may regard the elements of$G_n$as linear transformations on$\F_p^n$, […] • Find All the Eigenvalues of Power of Matrix and Inverse Matrix Let $A=\begin{bmatrix} 3 & -12 & 4 \\ -1 &0 &-2 \\ -1 & 5 & -1 \end{bmatrix}.$ Then find all eigenvalues of$A^5$. If$A$is invertible, then find all the eigenvalues of$A^{-1}\$.   Proof. We first determine all the eigenvalues of the matrix […]