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.
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.
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.
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 \(A\).
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.
Use a computer to find the least squares solutions of \(X b = y\) for the following matrices.
\(X = \begin{bmatrix} -1 & 2 \\ 2 & -3 \\ -1 & 3 \end{bmatrix}, ~ y = \begin{bmatrix} 4 \\ 1 \\ 2 \end{bmatrix}\)
\(X = \begin{bmatrix} 1 & -3 & - 3 \\ 1 & 5 & 1 \\ 1 & 7 & 2 \end{bmatrix}, ~ y = \begin{bmatrix} 5 \\ -3 \\ 5 \end{bmatrix}\)
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 |