# linear-algebra-eyecatch

by Yu · Published · Updated

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- 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, […]
- A Group Homomorphism that Factors though Another Group Let $G, H, K$ be groups. Let $f:G\to K$ be a group homomorphism and let $\pi:G\to H$ be a surjective group homomorphism such that the kernel of $\pi$ is included in the kernel of $f$: $\ker(\pi) \subset \ker(f)$. Define a map $\bar{f}:H\to K$ as follows. For each […]
- True or False Quiz About a System of Linear Equations (Purdue University Linear Algebra Exam) Which of the following statements are true? (a) A linear system of four equations in three unknowns is always inconsistent. (b) A linear system with fewer equations than unknowns must have infinitely many solutions. (c) […]
- Companion Matrix for a Polynomial Consider a polynomial \[p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\] where $a_i$ are real numbers. Define the matrix \[A=\begin{bmatrix} 0 & 0 & \dots & 0 &-a_0 \\ 1 & 0 & \dots & 0 & -a_1 \\ 0 & 1 & \dots & 0 & -a_2 \\ \vdots & […]
- 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) $\begin{bmatrix} -3 \\ 0 \\ 1 \\ 0 \end{bmatrix}$ […]
- Hyperplane in $n$-Dimensional Space Through Origin is a Subspace A hyperplane in $n$-dimensional vector space $\R^n$ is defined to be the set of vectors \[\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}\in \R^n\] satisfying the linear equation of the form \[a_1x_1+a_2x_2+\cdots+a_nx_n=b,\] […]
- If matrix product $AB$ is a square, then is $BA$ a square matrix? Let $A$ and $B$ are matrices such that the matrix product $AB$ is defined and $AB$ is a square matrix. Is it true that the matrix product $BA$ is also defined and $BA$ is a square matrix? If it is true, then prove it. If not, find a […]
- 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 , […]