# matrix-nonsingular-10seconds

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• The Sum of Subspaces is a Subspace of a Vector Space Let $V$ be a vector space over a field $K$. If $W_1$ and $W_2$ are subspaces of $V$, then prove that the subset $W_1+W_2:=\{\mathbf{x}+\mathbf{y} \mid \mathbf{x}\in W_1, \mathbf{y}\in W_2\}$ is a subspace of the vector space $V$.   Proof. We prove the […]
• Quiz 13 (Part 1) Diagonalize a Matrix Let $A=\begin{bmatrix} 2 & -1 & -1 \\ -1 &2 &-1 \\ -1 & -1 & 2 \end{bmatrix}.$ Determine whether the matrix $A$ is diagonalizable. If it is diagonalizable, then diagonalize $A$. That is, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that […]
• Is the Product of a Nilpotent Matrix and an Invertible Matrix Nilpotent? A square matrix $A$ is called nilpotent if there exists a positive integer $k$ such that $A^k=O$, where $O$ is the zero matrix. (a) If $A$ is a nilpotent $n \times n$ matrix and $B$ is an $n\times n$ matrix such that $AB=BA$. Show that the product $AB$ is nilpotent. (b) Let $P$ […]
• 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 […] • Isomorphism Criterion of Semidirect Product of Groups Let A, B be groups. Let \phi:B \to \Aut(A) be a group homomorphism. The semidirect product A \rtimes_{\phi} B with respect to \phi is a group whose underlying set is A \times B with group operation \[(a_1, b_1)\cdot (a_2, b_2)=(a_1\phi(b_1)(a_2), b_1b_2),$ where $a_i […] • How Many Solutions for$x+x=1$in a Ring? Is there a (not necessarily commutative) ring$R$with$1$such that the equation $x+x=1$ has more than one solutions$x\in R$? Solution. We claim that there is at most one solution$x$in the ring$R$. Suppose that we have two solutions$r, s \in R$. That is, we […] • Transpose of a Matrix and Eigenvalues and Related Questions Let$A$be an$n \times n$real matrix. Prove the followings. (a) The matrix$AA^{\trans}$is a symmetric matrix. (b) The set of eigenvalues of$A$and the set of eigenvalues of$A^{\trans}$are equal. (c) The matrix$AA^{\trans}$is non-negative definite. (An$n\times n$[…] • Prove that$\{ 1 , 1 + x , (1 + x)^2 \}$is a Basis for the Vector Space of Polynomials of Degree$2$or Less Let$\mathbf{P}_2$be the vector space of polynomials of degree$2$or less. (a) Prove that the set$\{ 1 , 1 + x , (1 + x)^2 \}$is a basis for$\mathbf{P}_2$. (b) Write the polynomial$f(x) = 2 + 3x - x^2$as a linear combination of the basis$\{ 1 , 1+x , (1+x)^2 […]

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