An Orthogonal Transformation from $\R^n$ to $\R^n$ is an Isomorphism
Let $\R^n$ be an inner product space with inner product $\langle \mathbf{x}, \mathbf{y}\rangle=\mathbf{x}^{\trans}\mathbf{y}$ for $\mathbf{x}, \mathbf{y}\in \R^n$.
A linear transformation $T:\R^n \to \R^n$ is called orthogonal transformation if for all $\mathbf{x}, \mathbf{y}\in […]
Every Plane Through the Origin in the Three Dimensional Space is a Subspace
Prove that every plane in the $3$-dimensional space $\R^3$ that passes through the origin is a subspace of $\R^3$.
Proof.
Each plane $P$ in $\R^3$ through the origin is given by the equation
\[ax+by+cz=0\]
for some real numbers $a, b, c$.
That is, the […]
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, […]
Matrix Representation of a Linear Transformation of the Vector Space $R^2$ to $R^2$
Let $B=\{\mathbf{v}_1, \mathbf{v}_2 \}$ be a basis for the vector space $\R^2$, and let $T:\R^2 \to \R^2$ be a linear transformation such that
\[T(\mathbf{v}_1)=\begin{bmatrix}
1 \\
-2
\end{bmatrix} \text{ and } T(\mathbf{v}_2)=\begin{bmatrix}
3 \\
1 […]
Short Exact Sequence and Finitely Generated Modules
Let $R$ be a ring with $1$. Let
\[0\to M\xrightarrow{f} M' \xrightarrow{g} M^{\prime\prime} \to 0 \tag{*}\]
be an exact sequence of left $R$-modules.
Prove that if $M$ and $M^{\prime\prime}$ are finitely generated, then $M'$ is also finitely generated.
[…]
Hyperplane in $n$-Dimensional Space Through Origin is a Subspace
A hyperplane in $n$-dimensional vector space $\R^n$ is defined to be the set of vectors
\[\begin{bmatrix}
x_1 \\
x_2 \\
\vdots \\
x_n
\end{bmatrix}\in \R^n\]
satisfying the linear equation of the form
\[a_1x_1+a_2x_2+\cdots+a_nx_n=b,\]
[…]
Sylow’s Theorem (Summary) In this post we review Sylow's theorem and as an example we solve the following problem.
Show that a group of order $200$ has a normal Sylow $5$-subgroup.
Review of Sylow's Theorem
One of the important theorems in group theory is Sylow's theorem.
Sylow's theorem is a […]
For Fixed Matrices $R, S$, the Matrices $RAS$ form a Subspace
Let $V$ be the vector space of $k \times k$ matrices. Then for fixed matrices $R, S \in V$, define the subset $W = \{ R A S \mid A \in V \}$.
Prove that $W$ is a vector subspace of $V$.
Proof.
We verify the subspace criteria: the zero vector of $V$ is in $W$, and […]