number2018

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Mathematics about the number 2018


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  • Does an Extra Vector Change the Span?Does an Extra Vector Change the Span? Suppose that a set of vectors $S_1=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is a spanning set of a subspace $V$ in $\R^5$. If $\mathbf{v}_4$ is another vector in $V$, then is the set \[S_2=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4\}\] still a spanning set for […]
  • A Linear Transformation $T: U\to V$ cannot be Injective if $\dim(U) > \dim(V)$A Linear Transformation $T: U\to V$ cannot be Injective if $\dim(U) > \dim(V)$ Let $U$ and $V$ be finite dimensional vector spaces over a scalar field $\F$. Consider a linear transformation $T:U\to V$. Prove that if $\dim(U) > \dim(V)$, then $T$ cannot be injective (one-to-one).   Hints. You may use the folowing facts. A linear […]
  • Dual Vector Space and Dual Basis, Some EqualityDual Vector Space and Dual Basis, Some Equality Let $V$ be a finite dimensional vector space over a field $k$ and let $V^*=\Hom(V, k)$ be the dual vector space of $V$. Let $\{v_i\}_{i=1}^n$ be a basis of $V$ and let $\{v^i\}_{i=1}^n$ be the dual basis of $V^*$. Then prove that \[x=\sum_{i=1}^nv^i(x)v_i\] for any vector $x\in […]
  • Eigenvalues of a Matrix and its Transpose are the SameEigenvalues of a Matrix and its Transpose are the Same Let $A$ be a square matrix. Prove that the eigenvalues of the transpose $A^{\trans}$ are the same as the eigenvalues of $A$.   Proof. Recall that the eigenvalues of a matrix are roots of its characteristic polynomial. Hence if the matrices $A$ and $A^{\trans}$ […]
  • In a Field of Positive Characteristic, $A^p=I$ Does Not Imply that $A$ is Diagonalizable.In a Field of Positive Characteristic, $A^p=I$ Does Not Imply that $A$ is Diagonalizable. Show that the matrix $A=\begin{bmatrix} 1 & \alpha\\ 0& 1 \end{bmatrix}$, where $\alpha$ is an element of a field $F$ of characteristic $p>0$ satisfies $A^p=I$ and the matrix is not diagonalizable over $F$ if $\alpha \neq 0$. Comment. Remark that if $A$ is a square […]
  • Vector Space of Polynomials and Coordinate VectorsVector Space of Polynomials and Coordinate Vectors Let $P_2$ be the vector space of all polynomials of degree two or less. Consider the subset in $P_2$ \[Q=\{ p_1(x), p_2(x), p_3(x), p_4(x)\},\] where \begin{align*} &p_1(x)=x^2+2x+1, &p_2(x)=2x^2+3x+1, \\ &p_3(x)=2x^2, &p_4(x)=2x^2+x+1. \end{align*} (a) Use the basis […]
  • Difference Between Ring Homomorphisms and Module HomomorphismsDifference Between Ring Homomorphisms and Module Homomorphisms Let $R$ be a ring with $1$ and consider $R$ as a module over itself. (a) Determine whether every module homomorphism $\phi:R\to R$ is a ring homomorphism. (b) Determine whether every ring homomorphism $\phi: R\to R$ is a module homomorphism. (c) If $\phi:R\to R$ is both a […]
  • Cosine and Sine Functions are Linearly IndependentCosine and Sine Functions are Linearly Independent Let $C[-\pi, \pi]$ be the vector space of all continuous functions defined on the interval $[-\pi, \pi]$. Show that the subset $\{\cos(x), \sin(x)\}$ in $C[-\pi, \pi]$ is linearly independent.   Proof. Note that the zero vector in the vector space $C[-\pi, \pi]$ is […]

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