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• 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 $\dim(U) > \dim(V)$ 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 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 $x\in […] • Eigenvalues of a Matrix and its Transpose are the Same Let$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$is Diagonalizable. Show that the matrix$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$is a square […] • Vector Space of Polynomials and Coordinate Vectors Let$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 Homomorphisms LetR$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 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 […]