# Stanford-university-exam-eye-catch

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- 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 […]
- Hyperplane Through Origin is Subspace of 4-Dimensional Vector Space Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix}$ satisfying \[2x+3y+5z+7w=0.\] Then prove that the set $S$ is a subspace of $\R^4$. (Linear Algebra Exam Problem, The Ohio State […]
- Similar Matrices Have the Same Eigenvalues Show that if $A$ and $B$ are similar matrices, then they have the same eigenvalues and their algebraic multiplicities are the same. Proof. We prove that $A$ and $B$ have the same characteristic polynomial. Then the result follows immediately since eigenvalues and algebraic […]
- Ascending Chain of Submodules and Union of its Submodules Let $R$ be a ring with $1$. Let $M$ be an $R$-module. Consider an ascending chain \[N_1 \subset N_2 \subset \cdots\] of submodules of $M$. Prove that the union \[\cup_{i=1}^{\infty} N_i\] is a submodule of $M$. Proof. To simplify the notation, let us […]
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