# module-theory-eye-catch

by Yu · Published · Updated

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- Linear Independent Vectors and the Vector Space Spanned By Them Let $V$ be a vector space over a field $K$. Let $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$ be linearly independent vectors in $V$. Let $U$ be the subspace of $V$ spanned by these vectors, that is, $U=\Span \{\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n\}$. Let […]
- Non-Prime Ideal of Continuous Functions Let $R$ be the ring of all continuous functions on the interval $[0,1]$. Let $I$ be the set of functions $f(x)$ in $R$ such that $f(1/2)=f(1/3)=0$. Show that the set $I$ is an ideal of $R$ but is not a prime ideal. Proof. We first show that $I$ is an ideal of […]
- The Matrix for the Linear Transformation of the Reflection Across a Line in the Plane Let $T:\R^2 \to \R^2$ be a linear transformation of the $2$-dimensional vector space $\R^2$ (the $x$-$y$-plane) to itself which is the reflection across a line $y=mx$ for some $m\in \R$. Then find the matrix representation of the linear transformation $T$ with respect to the […]
- Are these vectors in the Nullspace of the Matrix? Let $A=\begin{bmatrix} 1 & 0 & 3 & -2 \\ 0 &3 & 1 & 1 \\ 1 & 3 & 4 & -1 \end{bmatrix}$. For each of the following vectors, determine whether the vector is in the nullspace $\calN(A)$. (a) $\begin{bmatrix} -3 \\ 0 \\ 1 \\ 0 \end{bmatrix}$ […]
- Matrix Representations for Linear Transformations of the Vector Space of Polynomials Let $P_2(\R)$ be the vector space over $\R$ consisting of all polynomials with real coefficients of degree $2$ or less. Let $B=\{1,x,x^2\}$ be a basis of the vector space $P_2(\R)$. For each linear transformation $T:P_2(\R) \to P_2(\R)$ defined below, find the matrix representation […]
- The Quotient Ring $\Z[i]/I$ is Finite for a Nonzero Ideal of the Ring of Gaussian Integers Let $I$ be a nonzero ideal of the ring of Gaussian integers $\Z[i]$. Prove that the quotient ring $\Z[i]/I$ is finite. Proof. Recall that the ring of Gaussian integers is a Euclidean Domain with respect to the norm \[N(a+bi)=a^2+b^2\] for $a+bi\in \Z[i]$. In particular, […]
- Find all Column Vector $\mathbf{w}$ such that $\mathbf{v}\mathbf{w}=0$ for a Fixed Vector $\mathbf{v}$ Let $\mathbf{v} = \begin{bmatrix} 2 & -5 & -1 \end{bmatrix}$. Find all $3 \times 1$ column vectors $\mathbf{w}$ such that $\mathbf{v} \mathbf{w} = 0$. Solution. Let $\mathbf{w} = \begin{bmatrix} w_1 \\ w_2 \\ w_3 \end{bmatrix}$. Then we want \[\mathbf{v} […]
- Using Properties of Inverse Matrices, Simplify the Expression Let $A, B, C$ be $n\times n$ invertible matrices. When you simplify the expression \[C^{-1}(AB^{-1})^{-1}(CA^{-1})^{-1}C^2,\] which matrix do you get? (a) $A$ (b) $C^{-1}A^{-1}BC^{-1}AC^2$ (c) $B$ (d) $C^2$ (e) $C^{-1}BC$ (f) $C$ Solution. In this problem, we […]