Tagged: Ohio State.LA

A Linear Transformation from Vector Space over Rational Numbers to itself

Problem 75

Let $\Q$ denote the set of rational numbers (i.e., fractions of integers). Let $V$ denote the set of the form $x+y \sqrt{2}$ where $x,y \in \Q$. You may take for granted that the set $V$ is a vector space over the field $\Q$.

(a) Show that $B=\{1, \sqrt{2}\}$ is a basis for the vector space $V$ over $\Q$.

(b) Let $\alpha=a+b\sqrt{2} \in V$, and let $T_{\alpha}: V \to V$ be the map defined by
\[ T_{\alpha}(x+y\sqrt{2}):=(ax+2by)+(ay+bx)\sqrt{2}\in V\] for any $x+y\sqrt{2} \in V$.
Show that $T_{\alpha}$ is a linear transformation.

(c) Let $\begin{bmatrix}
x \\
y
\end{bmatrix}_B=x+y \sqrt{2}$.
Find the matrix $T_B$ such that
\[ T_{\alpha} (x+y \sqrt{2})=\left( T_B\begin{bmatrix}
x \\
y
\end{bmatrix}\right)_B,\] and compute $\det T_B$.

 

(The Ohio State University, Linear Algebra Exam)

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True or False: Eigenvalues of a Real Matrix Are Real Numbers

Problem 67

Answer the following questions regarding eigenvalues of a real matrix.

(a) True or False. If each entry of an $n \times n$ matrix $A$ is a real number, then the eigenvalues of $A$ are all real numbers.
(b) Find the eigenvalues of the matrix
\[B=\begin{bmatrix}
-2 & -1\\
5& 2
\end{bmatrix}.\]

(The Ohio State University, Linear Algebra Exam)

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Linear Independent Vectors, Invertible Matrix, and Expression of a Vector as a Linear Combinations

Problem 66

Consider the matrix
\[A=\begin{bmatrix}
1 & 2 & 1 \\
2 &5 &4 \\
1 & 1 & 0
\end{bmatrix}.\]


(a) Calculate the inverse matrix $A^{-1}$. If you think the matrix $A$ is not invertible, then explain why.


(b) Are the vectors
\[ \mathbf{A}_1=\begin{bmatrix}
1 \\
2 \\
1
\end{bmatrix}, \mathbf{A}_2=\begin{bmatrix}
2 \\
5 \\
1
\end{bmatrix},
\text{ and } \mathbf{A}_3=\begin{bmatrix}
1 \\
4 \\
0
\end{bmatrix}\] linearly independent?


(c) Write the vector $\mathbf{b}=\begin{bmatrix}
1 \\
1 \\
1
\end{bmatrix}$ as a linear combination of $\mathbf{A}_1$, $\mathbf{A}_2$, and $\mathbf{A}_3$.

(The Ohio State University, Linear Algebra Exam)

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Solving a System of Linear Equations By Using an Inverse Matrix

Problem 65

Consider the system of linear equations
\begin{align*}
x_1&= 2, \\
-2x_1 + x_2 &= 3, \\
5x_1-4x_2 +x_3 &= 2
\end{align*}

(a) Find the coefficient matrix and its inverse matrix.

(b) Using the inverse matrix, solve the system of linear equations.

(The Ohio State University, Linear Algebra Exam)

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