# nilpotent-matrix

• Positive definite Real Symmetric Matrix and its Eigenvalues A real symmetric $n \times n$ matrix $A$ is called positive definite if $\mathbf{x}^{\trans}A\mathbf{x}>0$ for all nonzero vectors $\mathbf{x}$ in $\R^n$. (a) Prove that the eigenvalues of a real symmetric positive-definite matrix $A$ are all positive. (b) Prove that if […]
• Complement of Independent Events are Independent Let $E$ and $F$ be independent events. Let $F^c$ be the complement of $F$. Prove that $E$ and $F^c$ are independent as well. Solution. Note that $E\cap F$ and $E \cap F^c$ are disjoint and $E = (E \cap F) \cup (E \cap F^c)$. It follows that P(E) = P(E \cap F) + P(E […] • A Condition that a Linear System has Nontrivial Solutions For what value(s) of a does the system have nontrivial solutions? \begin{align*} &x_1+2x_2+x_3=0\\ &-x_1-x_2+x_3=0\\ & 3x_1+4x_2+ax_3=0. \end{align*} Solution. First note that the system is homogeneous and hence it is consistent. Thus if the system has a nontrivial […] • 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 […] • A Linear Transformation T: U\to V cannot be Injective if \dim(U) > \dim(V) Let U and V be finite dimensional vector spaces over a scalar field \F. Consider a linear transformation T:U\to V. Prove that if \dim(U) > \dim(V), then T cannot be injective (one-to-one). Hints. You may use the folowing facts. A linear […] • Any Vector is a Linear Combination of Basis Vectors Uniquely Let B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\} be a basis for a vector space V over a scalar field K. Then show that any vector \mathbf{v}\in V can be written uniquely as \[\mathbf{v}=c_1\mathbf{v}_1+c_2\mathbf{v}_2+c_3\mathbf{v}_3, where $c_1, c_2, c_3$ are […]
• 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 […]
• Prove the Cauchy-Schwarz Inequality Let $\mathbf{a}, \mathbf{b}$ be vectors in $\R^n$. Prove the Cauchy-Schwarz inequality: $|\mathbf{a}\cdot \mathbf{b}|\leq \|\mathbf{a}\|\,\|\mathbf{b}\|.$   We give two proofs. Proof 1 Let $x$ be a variable and consider the length of the vector […]