Find a Basis of the Eigenspace Corresponding to a Given Eigenvalue
Let
\[A=\begin{bmatrix}
1 & 2 & 1 \\
-1 &4 &1 \\
2 & -4 & 0
\end{bmatrix}.\]
The matrix $A$ has an eigenvalue $2$.
Find a basis of the eigenspace $E_2$ corresponding to the eigenvalue $2$.
(The Ohio State University, Linear Algebra Final Exam […]
No Finite Abelian Group is Divisible
A nontrivial abelian group $A$ is called divisible if for each element $a\in A$ and each nonzero integer $k$, there is an element $x \in A$ such that $x^k=a$.
(Here the group operation of $A$ is written multiplicatively. In additive notation, the equation is written as $kx=a$.) That […]
Determine Whether There Exists a Nonsingular Matrix Satisfying $A^4=ABA^2+2A^3$
Determine whether there exists a nonsingular matrix $A$ if
\[A^4=ABA^2+2A^3,\]
where $B$ is the following matrix.
\[B=\begin{bmatrix}
-1 & 1 & -1 \\
0 &-1 &0 \\
2 & 1 & -4
\end{bmatrix}.\]
If such a nonsingular matrix $A$ exists, find the inverse […]
Determine a Value of Linear Transformation From $\R^3$ to $\R^2$
Let $T$ be a linear transformation from $\R^3$ to $\R^2$ such that
\[ T\left(\, \begin{bmatrix}
0 \\
1 \\
0
\end{bmatrix}\,\right) =\begin{bmatrix}
1 \\
2
\end{bmatrix} \text{ and }T\left(\, \begin{bmatrix}
0 \\
1 \\
1
[…]
Conditional Probability When the Sum of Two Geometric Random Variables Are Known
Let $X$ and $Y$ be geometric random variables with parameter $p$, with $0 \leq p \leq 1$. Assume that $X$ and $Y$ are independent.
Let $n$ be an integer greater than $1$. Let $k$ be a natural number with $k\leq n$. Then prove the formula
\[P(X=k \mid X + Y = n) = […]
Determine linear transformation using matrix representation
Let $T$ be the linear transformation from the $3$-dimensional vector space $\R^3$ to $\R^3$ itself satisfying the following relations.
\begin{align*}
T\left(\, \begin{bmatrix}
1 \\
1 \\
1
\end{bmatrix} \,\right)
=\begin{bmatrix}
1 \\
0 \\
1 […]
The Quotient by the Kernel Induces an Injective Homomorphism
Let $G$ and $G'$ be a group and let $\phi:G \to G'$ be a group homomorphism.
Show that $\phi$ induces an injective homomorphism from $G/\ker{\phi} \to G'$.
Outline.
Define $\tilde{\phi}([g])=\phi(g)$ and show that this is well-defined.
Show […]