MA 351 Fall 2026 (Aaron N. K. Yip)
Homework 1 Due: Thursday, Sept. 3, in class

Linear Algebra: Ideas and Applications, by Penney (4th edition) (Make sure you are using the 4th edition. You can access its online version from the Purdue Library page, using your Career account.)

p. 38 EXERCISES: 1.55(a,c,e,g,i,k), 1.59

Vector Algebra, by Johnston, Introduction to Linear and Matrix Algebra

(posted ref. in Week 1)

p. 9, 10 Exercises: 1.1.7, 1.1.8, 1.1.9, 1.1.10, 1.1.11

Beware that Johnston write the vectors as row vectors. To be consistent with this course, I want you to write every vector as column vectors.

Additional Problems:

  1. Consider the transformation of vectors from \( \mathbb{R}^2 \) to \( \mathbb{R}^2 \) defined by

    \[ T\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix} \]

    where \( a, b, c, d \) are some given numbers. Prove that \( T \) satisfies the following linearity properties (and hence \( T \) is legitimately called a linear transformation):

    1. \[ T\left( \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} + \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} \right) = T\begin{pmatrix} x_1 \\ x_2 \end{pmatrix} + T\begin{pmatrix} y_1 \\ y_2 \end{pmatrix} \]
    2. \[ T\left( \lambda \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \right) = \lambda T \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \]
    3. \[ T\left( \lambda \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} + \mu \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} \right) = \lambda T \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} + \mu T \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} \]

    (In the above, \( x_1, x_2, y_1, y_2, \lambda, \mu \) are arbitray numbers.)

  2. Now consider the following specific example of linear transformation \( T : \mathbb{R}^2 \longrightarrow \mathbb{R}^2 \):

    \[ T\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2x - y \\ x + 3y \end{pmatrix}. \]

    1. Consider the unit square in the \( xy \)-plane traced out by the points:

      \[ \begin{pmatrix} 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \end{pmatrix}. \]

      Find the image of these points under \( T \). (Connect the points by appropriate line segments.)

  1. Find the pre-image of \( \begin{pmatrix} p \\ q \end{pmatrix} \) under \( T \), i.e. find \( x, y \) such that \( T\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} p \\ q \end{pmatrix} \). You can express \( x, y \) in terms of \( p, q \).

  2. Consider the unit square in the \( pq \)-plane traced out by the points:

    \[ \begin{pmatrix} 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \end{pmatrix}. \]

    Find its pre-image in the \( xy \)-plane. (Again, connect the points by appropriate line segments.)