Math 422 Homework 3

Due Fri, Feb 21

Mon, Feb 10

  1. Consider an arbitrary quadrilateral with vertices \(a, b, c, d \in \mathbb{R}^2\). The blue segments connect the midpoints of adjacent sides. Use vectors to show that both blue segments have the same length and are parallel. Recall that the midpoint of two vectors \(x,y\) is \(\frac{1}{2}(x+y)\).
a b c d

In class we showed that for any vector \(x \in \mathbb{R}^n\), the vector \(x-\bar{x}\) can be calculated by using the formula \(x-\bar{x} = Ax\) where \(A = I - \frac{1}{n} ee^T\). Here \(I\) denotes the identity matrix in \(\mathbb{R}^{n \times n}\) and \(e \in \mathbb{R}^n\) is the vector with all 1 entries.

  1. An orthogonal projection is a matrix \(P\) in \(\mathbb{R}^{n \times n}\) that is (i) symmetric (i.e., \(P^T =P\)) and (ii) \(P^2 = P\). Prove that the matrix \(A = I - \frac{1}{n} ee^T\) is an orthogonal projection.

  2. We say that a vector \(y \in \mathbb{R}^m\) is in the range of a matrix \(A \in \mathbb{R}^{m \times n}\) if \(y = Ax\) for some \(x \in \mathbb{R}^n\). Recall that a vector \(v \in \mathbb{R}^n\) is an eigenvector of \(A \in \mathbb{R}^{n \times n}\) with eigenvalue \(\lambda \in \mathbb{R}\) if \(Av = \lambda v\). Prove that if \(y\) is in the range of an orthogonal projection \(P \in \mathbb{R}^{n \times n}\), then \(y\) is an eigenvector of \(P\) with eigenvalue 1.

  3. For any matrix \(A \in \mathbb{R}^{n \times n}\), the null space of \(A\) is the set of all vectors \(x \in \mathbb{R}^n\) such that \(Ax = 0\). If \(A\) is symmetric, prove that the null space of \(A\) is orthogonal to the range of \(A\).

Wed, Feb 12

  1. Write down and solve the normal equations to find the least squares regression line for the three points \((3,1)\), \((5,0)\), and \((-8,-1)\). Notice, this is not the same example we did in class since the \(y\) coordinates of the first two points are reversed (so the correlation is weaker).

Fri, Feb 14

  1. Prove that the linear transformation defined by a matrix \(A \in \mathbb{R}^{m \times n}\) is a 1-to-1 function if and only if \(\operatorname{Null}(A) = \{ 0 \}\). Hint: This is a two part proof. The first step is to prove that if \(A\) maps two different vectors \(u\) and \(v\) to the same \(y\), then \(\operatorname{Null(A)} \ne \{0\}\). Then prove that if \(A\) maps a non-zero vector to \(0\), then \(A\) is not 1-to-1.

  2. Use a computer to find the least squares solutions of \(X b = y\) for the following matrices.

  1. \(X = \begin{bmatrix} -1 & 2 \\ 2 & -3 \\ -1 & 3 \end{bmatrix}, ~ y = \begin{bmatrix} 4 \\ 1 \\ 2 \end{bmatrix}\)

  2. \(X = \begin{bmatrix} 1 & -3 & - 3 \\ 1 & 5 & 1 \\ 1 & 7 & 2 \end{bmatrix}, ~ y = \begin{bmatrix} 5 \\ -3 \\ 5 \end{bmatrix}\)

  1. Suppose an object is thrown off of a tall cliff (200 meters tall) and the elevation of the object above the ground is measured each second as it falls (see table below). Find the coefficients \(b_i\) of the parabola \(y = b_0 + b_1 t + b_2 t^2\). that best predicts the height of the object as a function of time. In particular, what is the initial vertical velocity and the acceleration of the object?
  Time (seconds)     Elevation (meters)  
0 200.0
1 205.3
2 200.2
3 185.5
4 161.5
5 127.3
6 83.5
7 29.8