# linear-transformation-eye-catch

by Yu · Published · Updated

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- 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) $\begin{bmatrix} -3 \\ 0 \\ 1 \\ 0 \end{bmatrix}$ […]
- Is the Map $T(f)(x) = (f(x))^2$ a Linear Transformation from the Vector Space of Real Functions? Let $C (\mathbb{R})$ be the vector space of real functions. Define the map $T$ by $T(f)(x) = (f(x))^2$ for $f \in C(\mathbb{R})$. Determine if $T$ is a linear transformation or not. If it is, determine the range of $T$. Solution. We claim that $T$ is not a […]
- Basis with Respect to Which the Matrix for Linear Transformation is Diagonal Let $P_1$ be the vector space of all real polynomials of degree $1$ or less. Consider the linear transformation $T: P_1 \to P_1$ defined by \[T(ax+b)=(3a+b)x+a+3,\] for any $ax+b\in P_1$. (a) With respect to the basis $B=\{1, x\}$, find the matrix of the linear transformation […]
- Idempotent Elements and Zero Divisors in a Ring and in an Integral Domain Prove the following statements. (a) If $a\neq 1$ is an idempotent element of $R$, then $a$ is a zero divisor. (b) Suppose that $R$ is an integral domain. Determine all the idempotent elements of $R$. Definitions (Idempotent, Zero Divisor, Integral […]
- Compute Power of Matrix If Eigenvalues and Eigenvectors Are Given Let $A$ be a $3\times 3$ matrix. Suppose that $A$ has eigenvalues $2$ and $-1$, and suppose that $\mathbf{u}$ and $\mathbf{v}$ are eigenvectors corresponding to $2$ and $-1$, respectively, where \[\mathbf{u}=\begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix} \text{ […]
- Conjugate of the Centralizer of a Set is the Centralizer of the Conjugate of the Set Let $X$ be a subset of a group $G$. Let $C_G(X)$ be the centralizer subgroup of $X$ in $G$. For any $g \in G$, show that $gC_G(X)g^{-1}=C_G(gXg^{-1})$. Proof. $(\subset)$ We first show that $gC_G(X)g^{-1} \subset C_G(gXg^{-1})$. Take any $h\in C_G(X)$. Then for […]
- The Cyclotomic Field of 8-th Roots of Unity is $\Q(\zeta_8)=\Q(i, \sqrt{2})$ Let $\zeta_8$ be a primitive $8$-th root of unity. Prove that the cyclotomic field $\Q(\zeta_8)$ of the $8$-th root of unity is the field $\Q(i, \sqrt{2})$. Proof. Recall that the extension degree of the cyclotomic field of $n$-th roots of unity is given by […]
- The Additive Group of Rational Numbers and The Multiplicative Group of Positive Rational Numbers are Not Isomorphic Let $(\Q, +)$ be the additive group of rational numbers and let $(\Q_{ > 0}, \times)$ be the multiplicative group of positive rational numbers. Prove that $(\Q, +)$ and $(\Q_{ > 0}, \times)$ are not isomorphic as groups. Proof. Suppose, towards a […]