# MIT-exam-eye-catch

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- Linear Dependent/Independent Vectors of Polynomials Let $p_1(x), p_2(x), p_3(x), p_4(x)$ be (real) polynomials of degree at most $3$. Which (if any) of the following two conditions is sufficient for the conclusion that these polynomials are linearly dependent? (a) At $1$ each of the polynomials has the value $0$. Namely $p_i(1)=0$ […]
- Use Coordinate Vectors to Show a Set is a Basis for the Vector Space of Polynomials of Degree 2 or Less Let $P_2$ be the vector space over $\R$ of all polynomials of degree $2$ or less. Let $S=\{p_1(x), p_2(x), p_3(x)\}$, where \[p_1(x)=x^2+1, \quad p_2(x)=6x^2+x+2, \quad p_3(x)=3x^2+x.\] (a) Use the basis $B=\{x^2, x, 1\}$ of $P_2$ to prove that the set $S$ is a basis for […]
- Nilpotent Matrices and Non-Singularity of Such Matrices Let $A$ be an $n \times n$ nilpotent matrix, that is, $A^m=O$ for some positive integer $m$, where $O$ is the $n \times n$ zero matrix. Prove that $A$ is a singular matrix and also prove that $I-A, I+A$ are both nonsingular matrices, where $I$ is the $n\times n$ identity […]
- If Two Matrices Have the Same Rank, Are They Row-Equivalent? If $A, B$ have the same rank, can we conclude that they are row-equivalent? If so, then prove it. If not, then provide a counterexample. Solution. Having the same rank does not mean they are row-equivalent. For a simple counterexample, consider $A = […]
- Explicit Field Isomorphism of Finite Fields (a) Let $f_1(x)$ and $f_2(x)$ be irreducible polynomials over a finite field $\F_p$, where $p$ is a prime number. Suppose that $f_1(x)$ and $f_2(x)$ have the same degrees. Then show that fields $\F_p[x]/(f_1(x))$ and $\F_p[x]/(f_2(x))$ are isomorphic. (b) Show that the polynomials […]
- Can $\Z$-Module Structure of Abelian Group Extend to $\Q$-Module Structure? If $M$ is a finite abelian group, then $M$ is naturally a $\Z$-module. Can this action be extended to make $M$ into a $\Q$-module? Proof. In general, we cannot extend a $\Z$-module into a $\Q$-module. We give a counterexample. Let $M=\Zmod{2}$ be the order […]
- Basis and Dimension of the Subspace of All Polynomials of Degree 4 or Less Satisfying Some Conditions. Let $P_4$ be the vector space consisting of all polynomials of degree $4$ or less with real number coefficients. Let $W$ be the subspace of $P_2$ by \[W=\{ p(x)\in P_4 \mid p(1)+p(-1)=0 \text{ and } p(2)+p(-2)=0 \}.\] Find a basis of the subspace $W$ and determine the dimension of […]
- Eigenvalues and Algebraic/Geometric Multiplicities of Matrix $A+cI$ Let $A$ be an $n \times n$ matrix and let $c$ be a complex number. (a) For each eigenvalue $\lambda$ of $A$, prove that $\lambda+c$ is an eigenvalue of the matrix $A+cI$, where $I$ is the identity matrix. What can you say about the eigenvectors corresponding to […]