Michigan-State-University-Abstract-Algebra-eye-catch
Michigan-State-University-Abstract-Algebra-eye-catch
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- If a Power of a Matrix is the Identity, then the Matrix is Diagonalizable
Let $A$ be an $n \times n$ complex matrix such that $A^k=I$, where $I$ is the $n \times n$ identity matrix.
Show that the matrix $A$ is diagonalizable.
Hint.
Use the fact that if the minimal polynomial for the matrix $A$ has distinct roots, then $A$ is […]
- Find All Matrices $B$ that Commutes With a Given Matrix $A$: $AB=BA$
Let
\[A=\begin{bmatrix}
1 & 3\\
2& 4
\end{bmatrix}.\]
Then
(a) Find all matrices
\[B=\begin{bmatrix}
x & y\\
z& w
\end{bmatrix}\]
such that $AB=BA$.
(b) Use the results of part (a) to exhibit $2\times 2$ matrices $B$ and $C$ such that
\[AB=BA \text{ and } […]
- A Matrix Similar to a Diagonalizable Matrix is Also Diagonalizable
Let $A, B$ be matrices. Show that if $A$ is diagonalizable and if $B$ is similar to $A$, then $B$ is diagonalizable.
Definitions/Hint.
Recall the relevant definitions.
Two matrices $A$ and $B$ are similar if there exists a nonsingular (invertible) matrix $S$ such […]
- A Group Homomorphism that Factors though Another Group
Let $G, H, K$ be groups. Let $f:G\to K$ be a group homomorphism and let $\pi:G\to H$ be a surjective group homomorphism such that the kernel of $\pi$ is included in the kernel of $f$: $\ker(\pi) \subset \ker(f)$.
Define a map $\bar{f}:H\to K$ as follows.
For each […]
- Beautiful Formulas for pi=3.14… The number $\pi$ is defined a s the ratio of a circle's circumference $C$ to its diameter $d$:
\[\pi=\frac{C}{d}.\]
$\pi$ in decimal starts with 3.14... and never end.
I will show you several beautiful formulas for $\pi$.
Art Museum of formulas for $\pi$ […]
- Three Linearly Independent Vectors in $\R^3$ Form a Basis. Three Vectors Spanning $\R^3$ Form a Basis.
Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a set of three-dimensional vectors in $\R^3$.
(a) Prove that if the set $B$ is linearly independent, then $B$ is a basis of the vector space $\R^3$.
(b) Prove that if the set $B$ spans $\R^3$, then $B$ is a basis of […]
- Union of Two Subgroups is Not a Group
Let $G$ be a group and let $H_1, H_2$ be subgroups of $G$ such that $H_1 \not \subset H_2$ and $H_2 \not \subset H_1$.
(a) Prove that the union $H_1 \cup H_2$ is never a subgroup in $G$.
(b) Prove that a group cannot be written as the union of two proper […]
- Determine the Values of $a$ so that $W_a$ is a Subspace
For what real values of $a$ is the set
\[W_a = \{ f \in C(\mathbb{R}) \mid f(0) = a \}\]
a subspace of the vector space $C(\mathbb{R})$ of all real-valued functions?
Solution.
The zero element of $C(\mathbb{R})$ is the function $\mathbf{0}$ defined by […]