# field-theory-eyecatch

by Yu · Published · Updated

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- Prove that $(A + B) \mathbf{v} = A\mathbf{v} + B\mathbf{v}$ Using the Matrix Components Let $A$ and $B$ be $n \times n$ matrices, and $\mathbf{v}$ an $n \times 1$ column vector. Use the matrix components to prove that $(A + B) \mathbf{v} = A\mathbf{v} + B\mathbf{v}$. Solution. We will use the matrix components $A = (a_{i j})_{1 \leq i, j \leq n}$, $B = […]
- 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 […]
- Calculate $A^{10}$ for a Given Matrix $A$ Find $A^{10}$, where $A=\begin{bmatrix} 4 & 3 & 0 & 0 \\ 3 &-4 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 \end{bmatrix}$. (Harvard University Exam) Solution. Let $B=\begin{bmatrix} 4 & 3\\ 3& -4 \end{bmatrix}$ […]
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- Vector Space of Polynomials and Coordinate Vectors Let $P_2$ be the vector space of all polynomials of degree two or less. Consider the subset in $P_2$ \[Q=\{ p_1(x), p_2(x), p_3(x), p_4(x)\},\] where \begin{align*} &p_1(x)=x^2+2x+1, &p_2(x)=2x^2+3x+1, \\ &p_3(x)=2x^2, &p_4(x)=2x^2+x+1. \end{align*} (a) Use the basis […]
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