linear=algebra-eye-catch2
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- Find the Inverse Linear Transformation if the Linear Transformation is an Isomorphism
Let $T:\R^3 \to \R^3$ be the linear transformation defined by the formula
\[T\left(\, \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} \,\right)=\begin{bmatrix}
x_1+3x_2-2x_3 \\
2x_1+3x_2 \\
x_2-x_3
\end{bmatrix}.\]
Determine whether $T$ is an […]
- Quiz 4: Inverse Matrix/ Nonsingular Matrix Satisfying a Relation
(a) Find the inverse matrix of
\[A=\begin{bmatrix}
1 & 0 & 1 \\
1 &0 &0 \\
2 & 1 & 1
\end{bmatrix}\]
if it exists. If you think there is no inverse matrix of $A$, then give a reason.
(b) Find a nonsingular $2\times 2$ matrix $A$ such that
\[A^3=A^2B-3A^2,\]
where […]
- Quiz 7. Find a Basis of the Range, Rank, and Nullity of a Matrix
(a) Let $A=\begin{bmatrix}
1 & 3 & 0 & 0 \\
1 &3 & 1 & 2 \\
1 & 3 & 1 & 2
\end{bmatrix}$.
Find a basis for the range $\calR(A)$ of $A$ that consists of columns of $A$.
(b) Find the rank and nullity of the matrix $A$ in part (a).
Solution.
(a) […]
- Example of an Infinite Group Whose Elements Have Finite Orders
Is it possible that each element of an infinite group has a finite order?
If so, give an example. Otherwise, prove the non-existence of such a group.
Solution.
We give an example of a group of infinite order each of whose elements has a finite order.
Consider […]
- Probability that Alice Wins n Games Before Bob Wins m Games
Alice and Bob play some game against each other. The probability that Alice wins one game is $p$. Assume that each game is independent.
If Alice wins $n$ games before Bob wins $m$ games, then Alice becomes the champion of the game. What is the probability that Alice becomes the […]
- Differentiating Linear Transformation is Nilpotent
Let $P_n$ be the vector space of all polynomials with real coefficients of degree $n$ or less.
Consider the differentiation linear transformation $T: P_n\to P_n$ defined by
\[T\left(\, f(x) \,\right)=\frac{d}{dx}f(x).\]
(a) Consider the case $n=2$. Let $B=\{1, x, x^2\}$ be a […]
- If the Order is an Even Perfect Number, then a Group is not Simple
(a) Show that if a group $G$ has the following order, then it is not simple.
$28$
$496$
$8128$
(b) Show that if the order of a group $G$ is equal to an even perfect number then the group is not simple.
Hint.
Use Sylow's theorem.
(See the post Sylow’s Theorem […]
- Find the Inverse Matrices if Matrices are Invertible by Elementary Row Operations
For each of the following $3\times 3$ matrices $A$, determine whether $A$ is invertible and find the inverse $A^{-1}$ if exists by computing the augmented matrix $[A|I]$, where $I$ is the $3\times 3$ identity matrix.
(a) $A=\begin{bmatrix}
1 & 3 & -2 \\
2 &3 &0 \\
[…]