# Prime-Ideal

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• 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 […]
• A Matrix Having One Positive Eigenvalue and One Negative Eigenvalue Prove that the matrix $A=\begin{bmatrix} 1 & 1.00001 & 1 \\ 1.00001 &1 &1.00001 \\ 1 & 1.00001 & 1 \end{bmatrix}$ has one positive eigenvalue and one negative eigenvalue. (University of California, Berkeley Qualifying Exam Problem)   Solution. Let us put […]
• 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, […]
• Find All Matrices Satisfying a Given Relation Let $a$ and $b$ be two distinct positive real numbers. Define matrices $A:=\begin{bmatrix} 0 & a\\ a & 0 \end{bmatrix}, \,\, B:=\begin{bmatrix} 0 & b\\ b& 0 \end{bmatrix}.$ Find all the pairs $(\lambda, X)$, where $\lambda$ is a real number and $X$ is a […]
• The Quotient by the Kernel Induces an Injective Homomorphism Let $G$ and $G'$ be a group and let $\phi:G \to G'$ be a group homomorphism.  Show that $\phi$ induces an injective homomorphism from $G/\ker{\phi} \to G'$.   Outline. Define $\tilde{\phi}([g])=\phi(g)$ and show that this is well-defined. Show […]
• Differentiating Linear Transformation is Nilpotent Let $P_n$ be the vector space of all polynomials with real coefficients of degree $n$ or less. Consider the differentiation linear transformation $T: P_n\to P_n$ defined by $T\left(\, f(x) \,\right)=\frac{d}{dx}f(x).$ (a) Consider the case $n=2$. Let $B=\{1, x, x^2\}$ be a […]
• If Quotient $G/H$ is Abelian Group and $H < K \triangleleft G$, then $G/K$ is Abelian Let $H$ and $K$ be normal subgroups of a group $G$. Suppose that $H < K$ and the quotient group $G/H$ is abelian. Then prove that $G/K$ is also an abelian group.   Solution. We will give two proofs. Hint (The third isomorphism theorem) Recall the third […]
• Find Bases for the Null Space, Range, and the Row Space of a $5\times 4$ Matrix Let $A=\begin{bmatrix} 1 & -1 & 0 & 0 \\ 0 &1 & 1 & 1 \\ 1 & -1 & 0 & 0 \\ 0 & 2 & 2 & 2\\ 0 & 0 & 0 & 0 \end{bmatrix}.$ (a) Find a basis for the null space $\calN(A)$. (b) Find a basis of the range $\calR(A)$. (c) Find a basis of the […]