# Prime-Ideal

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• Express a Vector as a Linear Combination of Other Vectors Express the vector $\mathbf{b}=\begin{bmatrix} 2 \\ 13 \\ 6 \end{bmatrix}$ as a linear combination of the vectors $\mathbf{v}_1=\begin{bmatrix} 1 \\ 5 \\ -1 \end{bmatrix}, \mathbf{v}_2= \begin{bmatrix} 1 \\ 2 \\ 1 […] • If the Quotient Ring is a Field, then the Ideal is Maximal Let R be a ring with unit 1\neq 0. Prove that if M is an ideal of R such that R/M is a field, then M is a maximal ideal of R. (Do not assume that the ring R is commutative.) Proof. Let I be an ideal of R such that \[M \subset I \subset […] • Find the Formula for the Power of a Matrix Using Linear Recurrence Relation Suppose that A is 2\times 2 matrix that has eigenvalues -1 and 3. Then for each positive integer n find a_n and b_n such that \[A^{n+1}=a_nA+b_nI,$ where $I$ is the $2\times 2$ identity matrix.   Solution. Since $-1, 3$ are eigenvalues of the […]
• Matrix Operations with Transpose Calculate the following expressions, using the following matrices: $A = \begin{bmatrix} 2 & 3 \\ -5 & 1 \end{bmatrix}, \qquad B = \begin{bmatrix} 0 & -1 \\ 1 & -1 \end{bmatrix}, \qquad \mathbf{v} = \begin{bmatrix} 2 \\ -4 \end{bmatrix}$ (a) $A B^\trans + \mathbf{v} […] • The Range and Nullspace of the Linear Transformation$T (f) (x) = x f(x)$For an integer$n > 0$, let$\mathrm{P}_n$be the vector space of polynomials of degree at most$n$. The set$B = \{ 1 , x , x^2 , \cdots , x^n \}$is a basis of$\mathrm{P}_n$, called the standard basis. Let$T : \mathrm{P}_n \rightarrow \mathrm{P}_{n+1}$be the map defined by, […] • Similar Matrices Have the Same Eigenvalues Show that if$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$have the same characteristic polynomial. Then the result follows immediately since eigenvalues and algebraic […] • Powers of a Diagonal Matrix Let$A=\begin{bmatrix} a & 0\\ 0& b \end{bmatrix}$. Show that (1)$A^n=\begin{bmatrix} a^n & 0\\ 0& b^n \end{bmatrix}$for any$n \in \N$. (2) Let$B=S^{-1}AS$, where$S$be an invertible$2 \times 2$matrix. Show that$B^n=S^{-1}A^n S$for any$n \in […]
• The Coordinate Vector for a Polynomial with respect to the Given Basis Let $\mathrm{P}_3$ denote the set of polynomials of degree $3$ or less with real coefficients. Consider the ordered basis $B = \left\{ 1+x , 1+x^2 , x - x^2 + 2x^3 , 1 - x - x^2 \right\}.$ Write the coordinate vector for the polynomial $f(x) = -3 + 2x^3$ in terms of the basis […]