# Kyushu-University-Linear-Algebra

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• Inverse Matrix of Positive-Definite Symmetric Matrix is Positive-Definite Suppose $A$ is a positive definite symmetric $n\times n$ matrix. (a) Prove that $A$ is invertible. (b) Prove that $A^{-1}$ is symmetric. (c) Prove that $A^{-1}$ is positive-definite. (MIT, Linear Algebra Exam Problem)   Proof. (a) Prove that $A$ is […]
• Every Prime Ideal is Maximal if $a^n=a$ for any Element $a$ in the Commutative Ring Let $R$ be a commutative ring with identity $1\neq 0$. Suppose that for each element $a\in R$, there exists an integer $n > 1$ depending on $a$. Then prove that every prime ideal is a maximal ideal.   Hint. Let $R$ be a commutative ring with $1$ and $I$ be an ideal […]
• Idempotent Linear Transformation and Direct Sum of Image and Kernel Let $A$ be the matrix for a linear transformation $T:\R^n \to \R^n$ with respect to the standard basis of $\R^n$. We assume that $A$ is idempotent, that is, $A^2=A$. Then prove that $\R^n=\im(T) \oplus \ker(T).$   Proof. To prove the equality $\R^n=\im(T) […] • The Additive Group of Rational Numbers and The Multiplicative Group of Positive Rational Numbers are Not Isomorphic Let$(\Q, +)$be the additive group of rational numbers and let$(\Q_{ > 0}, \times)$be the multiplicative group of positive rational numbers. Prove that$(\Q, +)$and$(\Q_{ > 0}, \times)$are not isomorphic as groups. Proof. Suppose, towards a […] • Find a Basis for the Subspace spanned by Five Vectors Let$S=\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4},\mathbf{v}_{5}\}$where $\mathbf{v}_{1}= \begin{bmatrix} 1 \\ 2 \\ 2 \\ -1 \end{bmatrix} ,\;\mathbf{v}_{2}= \begin{bmatrix} 1 \\ 3 \\ 1 \\ 1 \end{bmatrix} ,\;\mathbf{v}_{3}= \begin{bmatrix} 1 \\ 5 \\ -1 […] • Matrix Operations with Transpose Calculate the following expressions, using the following matrices: \[A = \begin{bmatrix} 2 & 3 \\ -5 & 1 \end{bmatrix}, \qquad B = \begin{bmatrix} 0 & -1 \\ 1 & -1 \end{bmatrix}, \qquad \mathbf{v} = \begin{bmatrix} 2 \\ -4 \end{bmatrix}$ (a)$A B^\trans + \mathbf{v} […]
• Two Matrices are Nonsingular if and only if the Product is Nonsingular An $n\times n$ matrix $A$ is called nonsingular if the only vector $\mathbf{x}\in \R^n$ satisfying the equation $A\mathbf{x}=\mathbf{0}$ is $\mathbf{x}=\mathbf{0}$. Using the definition of a nonsingular matrix, prove the following statements. (a) If $A$ and $B$ are $n\times […] • Idempotent Matrices are Diagonalizable Let$A$be an$n\times n$idempotent matrix, that is,$A^2=A$. Then prove that$A$is diagonalizable. We give three proofs of this problem. The first one proves that$\R^n$is a direct sum of eigenspaces of$A$, hence$A\$ is diagonalizable. The second proof proves […]