Author: Yu

The Additive Group $\R$ is Isomorphic to the Multiplicative Group $\R^{+}$ by Exponent Function

Problem 322

Let $\R=(\R, +)$ be the additive group of real numbers and let $\R^{\times}=(\R\setminus\{0\}, \cdot)$ be the multiplicative group of real numbers.

(a) Prove that the map $\exp:\R \to \R^{\times}$ defined by
\[\exp(x)=e^x\] is an injective group homomorphism.

(b) Prove that the additive group $\R$ is isomorphic to the multiplicative group
\[\R^{+}=\{x \in \R \mid x > 0\}.\]

 
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Solve a Linear Recurrence Relation Using Vector Space Technique

Problem 321

Let $V$ be a real vector space of all real sequences
\[(a_i)_{i=1}^{\infty}=(a_1, a_2, \dots).\] Let $U$ be a subspace of $V$ defined by
\[U=\{(a_i)_{i=1}^{\infty}\in V \mid a_{n+2}=2a_{n+1}+3a_{n} \text{ for } n=1, 2,\dots \}.\] Let $T$ be the linear transformation from $U$ to $U$ defined by
\[T\big((a_1, a_2, \dots)\big)=(a_2, a_3, \dots). \]

(a) Find the eigenvalues and eigenvectors of the linear transformation $T$.

(b) Use the result of (a), find a sequence $(a_i)_{i=1}^{\infty}$ satisfying $a_1=2, a_2=7$.

 
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Problems and Solutions About Similar Matrices

Problem 319

Let $A, B$, and $C$ be $n \times n$ matrices and $I$ be the $n\times n$ identity matrix.
Prove the following statements.

(a) If $A$ is similar to $B$, then $B$ is similar to $A$.

(b) $A$ is similar to itself.

(c) If $A$ is similar to $B$ and $B$ is similar to $C$, then $A$ is similar to $C$.

(d) If $A$ is similar to the identity matrix $I$, then $A=I$.

(e) If $A$ or $B$ is nonsingular, then $AB$ is similar to $BA$.

(f) If $A$ is similar to $B$, then $A^k$ is similar to $B^k$ for any positive integer $k$.

 
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If Column Vectors Form Orthonormal set, is Row Vectors Form Orthonormal Set?

Problem 317

Suppose that $A$ is a real $n\times n$ matrix.

(a) Is it true that $A$ must commute with its transpose?

(b) Suppose that the columns of $A$ (considered as vectors) form an orthonormal set.
Is it true that the rows of $A$ must also form an orthonormal set?

(University of California, Berkeley, Linear Algebra Qualifying Exam)

 
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Is there an Odd Matrix Whose Square is $-I$?

Problem 316

Let $n$ be an odd positive integer.
Determine whether there exists an $n \times n$ real matrix $A$ such that
\[A^2+I=O,\] where $I$ is the $n \times n$ identity matrix and $O$ is the $n \times n$ zero matrix.

If such a matrix $A$ exists, find an example. If not, prove that there is no such $A$.

How about when $n$ is an even positive number?

 
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Basis with Respect to Which the Matrix for Linear Transformation is Diagonal

Problem 315

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 $T$.

(b) Find a basis $B’$ of the vector space $P_1$ such that the matrix of $T$ with respect to $B’$ is a diagonal matrix.

(c) Express $f(x)=5x+3$ as a linear combination of basis vectors of $B’$.

 
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Matrix of Linear Transformation with respect to a Basis Consisting of Eigenvectors

Problem 314

Let $T$ be the linear transformation from the vector space $\R^2$ to $\R^2$ itself given by
\[T\left( \begin{bmatrix}
x_1 \\
x_2
\end{bmatrix} \right)= \begin{bmatrix}
3x_1+x_2 \\
x_1+3x_2
\end{bmatrix}.\]

(a) Verify that the vectors
\[\mathbf{v}_1=\begin{bmatrix}
1 \\
-1
\end{bmatrix} \text{ and } \mathbf{v}_2=\begin{bmatrix}
1 \\
1
\end{bmatrix}\] are eigenvectors of the linear transformation $T$, and conclude that $B=\{\mathbf{v}_1, \mathbf{v}_2\}$ is a basis of $\R^2$ consisting of eigenvectors.

(b) Find the matrix of $T$ with respect to the basis $B=\{\mathbf{v}_1, \mathbf{v}_2\}$.

 
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Quiz 6. Determine Vectors in Null Space, Range / Find a Basis of Null Space

Problem 313

(a) Let $A=\begin{bmatrix}
1 & 2 & 1 \\
3 &6 &4
\end{bmatrix}$ and let
\[\mathbf{a}=\begin{bmatrix}
-3 \\
1 \\
1
\end{bmatrix}, \qquad \mathbf{b}=\begin{bmatrix}
-2 \\
1 \\
0
\end{bmatrix}, \qquad \mathbf{c}=\begin{bmatrix}
1 \\
1
\end{bmatrix}.\] For each of the vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$, determine whether the vector is in the null space $\calN(A)$. Do the same for the range $\calR(A)$.

(b) Find a basis of the null space of the matrix $B=\begin{bmatrix}
1 & 1 & 2 \\
-2 &-2 &-4
\end{bmatrix}$.

 
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Find a Condition that a Vector be a Linear Combination

Problem 312

Let
\[\mathbf{v}=\begin{bmatrix}
a \\
b \\
c
\end{bmatrix}, \qquad \mathbf{v}_1=\begin{bmatrix}
1 \\
2 \\
0
\end{bmatrix}, \qquad \mathbf{v}_2=\begin{bmatrix}
2 \\
-1 \\
2
\end{bmatrix}.\] Find the necessary and sufficient condition so that the vector $\mathbf{v}$ is a linear combination of the vectors $\mathbf{v}_1, \mathbf{v}_2$.

 
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Solve Linear Recurrence Relation Using Linear Algebra (Eigenvalues and Eigenvectors)

Problem 310

Let $V$ be a real vector space of all real sequences
\[(a_i)_{i=1}^{\infty}=(a_1, a_2, \dots).\] Let $U$ be the subspace of $V$ consisting of all real sequences that satisfy the linear recurrence relation
\[a_{k+2}-5a_{k+1}+3a_{k}=0\] for $k=1, 2, \dots$.
Let $T$ be the linear transformation from $U$ to $U$ defined by
\[T\big((a_1, a_2, \dots)\big)=(a_2, a_3, \dots). \]

Let $B=\{\mathbf{u}_1, \mathbf{u}_2\}$ be a basis of $U$, where
\begin{align*}
\mathbf{u}_1&=(1, 0, -3, -15, -66, \dots)\\
\mathbf{u}_2&=(0, 1, 5, 22, 95, \dots).
\end{align*}
Let $A$ be the matrix representation of the linear transformation $T: U \to U$ with respect to the basis $B$.

(a) Find the eigenvalues and eigenvectors of $T$.

(b) Use the result of (a), find a sequence $(a_i)_{i=1}^{\infty}$ satisfying the linear recurrence relation $a_{k+2}-5a_{k+1}+3a_{k}=0$ and the initial condition $a_1=1, a_2=1$.

(c) Find the formula for the sequences $(a_i)_{i=1}^{\infty}$ satisfying the linear recurrence relation $a_{k+2}-5a_{k+1}+3a_{k}=0$ and express it using $a_1, a_2$.

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Matrix Representation of a Linear Transformation of Subspace of Sequences Satisfying Recurrence Relation

Problem 309

Let $V$ be a real vector space of all real sequences
\[(a_i)_{i=1}^{\infty}=(a_1, a_2, \dots).\] Let $U$ be the subspace of $V$ consisting of all real sequences that satisfy the linear recurrence relation $a_{k+2}-5a_{k+1}+3a_{k}=0$ for $k=1, 2, \dots$.

(a) Let
\begin{align*}
\mathbf{u}_1&=(1, 0, -3, -15, -66, \dots)\\
\mathbf{u}_2&=(0, 1, 5, 22, 95, \dots)
\end{align*}
be vectors in $U$. Prove that $\{\mathbf{u}_1, \mathbf{u}_2\}$ is a basis of $U$ and conclude that the dimension of $U$ is $2$.


(b) Let $T$ be a map from $U$ to $U$ defined by
\[T\big((a_1, a_2, \dots)\big)=(a_2, a_3, \dots). \] Verify that the map $T$ actually sends a vector $(a_i)_{i=1}^{\infty}\in V$ to a vector $T\big((a_i)_{i=1}^{\infty}\big)$ in $U$, and show that $T$ is a linear transformation from $U$ to $U$.


(c) With respect to the basis $\{\mathbf{u}_1, \mathbf{u}_2\}$ obtained in (a), find the matrix representation $A$ of the linear transformation $T:U \to U$ from (b).

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Torsion Subgroup of an Abelian Group, Quotient is a Torsion-Free Abelian Group

Problem 307

Let $A$ be an abelian group and let $T(A)$ denote the set of elements of $A$ that have finite order.

(a) Prove that $T(A)$ is a subgroup of $A$.

(The subgroup $T(A)$ is called the torsion subgroup of the abelian group $A$ and elements of $T(A)$ are called torsion elements.)

(b) Prove that the quotient group $G=A/T(A)$ is a torsion-free abelian group. That is, the only element of $G$ that has finite order is the identity element.

 
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Quiz 5: Example and Non-Example of Subspaces in 3-Dimensional Space

Problem 304

Problem 1 Let $W$ be the subset of the $3$-dimensional vector space $\R^3$ defined by
\[W=\left\{ \mathbf{x}=\begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix}\in \R^3 \quad \middle| \quad 2x_1x_2=x_3 \right\}.\]

(a) Which of the following vectors are in the subset $W$? Choose all vectors that belong to $W$.
\[(1) \begin{bmatrix}
0 \\
0 \\
0
\end{bmatrix} \qquad(2) \begin{bmatrix}
1 \\
2 \\
2
\end{bmatrix} \qquad(3)\begin{bmatrix}
3 \\
0 \\
0
\end{bmatrix} \qquad(4) \begin{bmatrix}
0 \\
0
\end{bmatrix} \qquad(5) \begin{bmatrix}
1 & 2 & 4 \\
1 &2 &4
\end{bmatrix} \qquad(6) \begin{bmatrix}
1 \\
-1 \\
-2
\end{bmatrix}.\]

(b) Determine whether $W$ is a subspace of $\R^3$ or not.
 


Problem 2 Let $W$ be the subset of $\R^3$ defined by
\[W=\left\{ \mathbf{x}=\begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} \in \R^3 \quad \middle| \quad x_1=3x_2 \text{ and } x_3=0 \right\}.\] Determine whether the subset $W$ is a subspace of $\R^3$ or not.

 
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