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• 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 $B=\{\sin^2(x), \cos^2(x)\}$ […]
• For Fixed Matrices $R, S$, the Matrices $RAS$ form a Subspace Let $V$ be the vector space of $k \times k$ matrices. Then for fixed matrices $R, S \in V$, define the subset $W = \{ R A S \mid A \in V \}$. Prove that $W$ is a vector subspace of $V$.   Proof. We verify the subspace criteria: the zero vector of $V$ is in $W$, and […]
• Find a Matrix so that a Given Subset is the Null Space of the Matrix, hence it’s a Subspace Let $W$ be the subset of $\R^3$ defined by \[W=\left \{ \mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\in \R^3 \quad \middle| \quad 5x_1-2x_2+x_3=0 \right \}.\] Exhibit a $1\times 3$ matrix $A$ such that $W=\calN(A)$, the null space of $A$. […]
• Does an Extra Vector Change the Span? Suppose that a set of vectors $S_1=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is a spanning set of a subspace $V$ in $\R^5$. If $\mathbf{v}_4$ is another vector in $V$, then is the set \[S_2=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4\}\] still a spanning set for […]
• Three Equivalent Conditions for a Ring to be a Field Let $R$ be a ring with $1$. Prove that the following three statements are equivalent. The ring $R$ is a field. The only ideals of $R$ are $(0)$ and $R$. Let $S$ be any ring with $1$. Then any ring homomorphism $f:R \to S$ is injective.   Proof. […]
• Given the Data of Eigenvalues, Determine if the Matrix is Invertible In each of the following cases, can we conclude that $A$ is invertible? If so, find an expression for $A^{-1}$ as a linear combination of positive powers of $A$. If $A$ is not invertible, explain why not. (a) The matrix $A$ is a $3 \times 3$ matrix with eigenvalues $\lambda=i , […]
• The Polynomial Rings $\Z[x]$ and $\Q[x]$ are Not Isomorphic Prove that the rings $\Z[x]$ and $\Q[x]$ are not isomoprhic.   Proof. We give three proofs. The first two proofs use only the properties of ring homomorphism. The third proof resort to the units of rings. If you are familiar with units of $\Z[x]$, then the […]
• Eigenvalues of Orthogonal Matrices Have Length 1. Every $3\times 3$ Orthogonal Matrix Has 1 as an Eigenvalue (a) Let $A$ be a real orthogonal $n\times n$ matrix. Prove that the length (magnitude) of each eigenvalue of $A$ is $1$. (b) Let $A$ be a real orthogonal $3\times 3$ matrix and suppose that the determinant of $A$ is $1$. Then prove that $A$ has $1$ as an […]