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Tagged: Ohio State.LA
by Yu · Published 08/18/2016 · Last modified 11/17/2017
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 $\begin{bmatrix}
x \\
\end{bmatrix}_B=x+y \sqrt{2}$.
Find the matrix $T_B$ such that
\[ T_{\alpha} (x+y \sqrt{2})=\left( T_B\begin{bmatrix}
\end{bmatrix}\right)_B,\] 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
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
\[B=\begin{bmatrix}
-2 & -1\\
5& 2
\end{bmatrix}.\]
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Linear Independent Vectors, Invertible Matrix, and Expression of a Vector as a Linear Combinations
Consider the matrix
\[A=\begin{bmatrix}
1 & 2 & 1 \\
2 &5 &4 \\
1 & 1 & 0
(a) Calculate the inverse matrix $A^{-1}$. If you think the matrix $A$ is not invertible, then explain why.
(b) Are the vectors
\[ \mathbf{A}_1=\begin{bmatrix}
1 \\
\end{bmatrix}, \mathbf{A}_2=\begin{bmatrix}
\end{bmatrix},
\text{ and } \mathbf{A}_3=\begin{bmatrix}
\end{bmatrix}\] linearly independent?
(c) Write the vector $\mathbf{b}=\begin{bmatrix}
\end{bmatrix}$ as a linear combination of $\mathbf{A}_1$, $\mathbf{A}_2$, and $\mathbf{A}_3$.
Solving a System of Linear Equations By Using an Inverse Matrix
Consider the system of linear equations
x_1&= 2, \\
-2x_1 + x_2 &= 3, \\
5x_1-4x_2 +x_3 &= 2
(a) Find the coefficient matrix and its inverse matrix.
(b) Using the inverse matrix, solve the system of linear equations.
How to Calculate and Simplify a Matrix Polynomial
Let $T=\begin{bmatrix}
\end{bmatrix}$.
Calculate and simplify the expression
\[-T^3+4T^2+5T-2I,\] where $I$ is the $3\times 3$ identity matrix.
(The Ohio State University Linear Algebra Exam)
A Linear Transformation Maps the Zero Vector to the Zero Vector
Let $T : \mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation.
Let $\mathbf{0}_n$ and $\mathbf{0}_m$ be zero vectors of $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively.
Show that $T(\mathbf{0}_n)=\mathbf{0}_m$.
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