Problems

01

Gram-Schmidt calculation

Perform Gram-Schmidt to find an orthogonal basis for the span of the column vectors of the following matrix:

$$\left(\begin{array}{cccc}-10& 13& 7& -11\\ 2& 1& -5& 3\\ -6& 3& 13& -3\\ 16& -16& -2& 5\\ 2& 1& -5& -7\end{array}\right)$$

02

Distance point to plane

How far is the point $\mathbf{x}=(1,2,3,4)$ from the span:

$$⟨\left(\begin{array}{c}1\\ -1\\ 0\\ 0\end{array}\right),\left(\begin{array}{c}1\\ 1\\ 1\\ 2\end{array}\right)⟩?$$

03

Decomposition into subspace and its complement

Suppose $V\subset {\mathbb{R}}^n$ is a subspace. Prove that every vector $\mathbf{x}\in {\mathbb{R}}^n$ can be written uniquely as a sum:

$$\mathbf{x}=\mathbf{v}+\mathbf{w}$$

such that $\mathbf{v}\in V$ and $\mathbf{w}$ is perpendicular to $V$ (i.e. perpendicular to everything in $V$).

(Hint: the standard way to show an item is unique is to posit two such items with the specified property, and show that they are equal to each other.)

04

Best fit line, nonzero averages

Find the best fit line for these five data points, and compute the correlation coefficient:

$$(-5,-5),(-4,3)(-3,1)(-2,-3)(-1,4).$$