18.06 Problem Set 6

Due Friday, October 21 at 11am.

Problem 1 (5+5+5+5 points)

(a) If $P$ is the projection matrix onto the null space of $A$, then $Py − y$, for any $y$, is in the __________ space of $A$.

(b) If $Ax = b$ has a solution $x$, then the closest vector to $b$ in $N(A^T)$ is __________. (Try drawing a picture.)

(c) If the rows of $A$ (an $m \times n$ matrix) are independent, then the dimension $N(A^T A)$ is __________.

(d) If a matrix $U$ has orthonormal rows, then $I = \_\_\_\_\_\_\_\_\_\_$ and the projection matrix onto the row space of U is __________. (Your answers should be simplified expressions involving $U$ and $U^T$ only.)

Problem 2 (5+10 points)

In class, we saw that the orthogonal projection $p$ of a vector $b$ onto $C(A)$ is given by $p = A\hat{x}$ where $\hat{x}$ is a solution to the "normal equations" $A^T A \hat{x} = A^T b$. We showed in class that these equations are always solvable, because $A^T b \in C(A^T) = C(A^T A)$.

(a) The least-square solution $\hat{x}$ is unique if $A$ is __________, in which case $A^T A$ is __________.

(b) The least-square solution $\hat{x}$ is not unique if $A$ is __________, in which case $A^T A$ is __________. However, the projection $p = A\hat{x}$ is still unique: if you have two solutions $\hat{x}$ and $\hat{x}'$ to the normal equations, $A\hat{x} - A\hat{x}' = \_\_\_\_\_\_\_\_$ because __________.

Problem 3 (5+5+10 points)

In this problem you will fit the motion of a projectile to a parabola in order to extract its velocity and the Earth's gravitational acceleration g.

If you launch a mass into the air from some position $x(0), y(0)$ with initial velocity $v_x(0), v_y(0)$, and friction is negligible, then you probably remember from 8.01 or high-school physics that the trajectory is a parabola:

$$ x(t) = x(0) + v_x(0) t $$$$ y(t) = \underbrace{y(0) + v_y(0) t - \frac{g}{2} t^2}_{\mbox{parabola }y(t)} = \underbrace{y(0) + \frac{v_y(0)}{v_x(0)} [x - x(0)] - \frac{g}{2v_x(0)^2} [x - x(0)]^2}_{\mbox{parabola }y(x)} $$

The textbook Physics2000 by E. R. Huggins contains this strobe photograph of a steel ball launched from the left, with a picture taken every 0.1 seconds. The spatial grid in the background is numbered in 10s of cm (so that the numbers "1,2,3,…,10" refer to 10cm,20cm,30cm,100cm, and the fine grid lines are 1cm).

(a) Extract the data from the image for the 6 locations of the ball, and define Julia arrays x = [...], y = [...], and t = [...] holding the coordinates $x,y,t$ (horizontal,vertical,time) of each ball location in meters and seconds, respectively. You can just zoom in and read the coordinates off the image by counting the fine grid lines, or use a fancier tool like WebPlotDigitizer. Plot the data $x(t)$ and $y(t)$.

(b) Perform a least-square fit to $x(0) + v_x(0) t$ to extract the velocity $v_x(0)$. That is, define a $6\times 2$ matrix X so that the least-square solution X \ x returns the 2-component vector $[x(0), v_x(0)]$ of the unknowns. Plot your linear fit along with the data $x(t)$.

(c) Perform a least-square fit to $y(0) + v_y(0) t - \frac{g}{2} t^2$ to extract the Earth's gravitational constant $g$ in m/s² (you should get roughly 9.8m/s²). That is, define a $6\times 3$ matrix Y so that the least-square solution Y \ y returns the 3-component vector $[y(0), v_y(0), g]$ of the unknowns. Plot your parabolic fit along with the data $y(t)$.

A code outline is given below; fill in the blanks:

# part (a)

x = [...] # FILL IN
y = [...] # FILL IN
t = [...] # FILL IN

using PyPlot
plot(t, x, "bo-")
plot(t, y, "r*-")
xlabel("time (seconds)")
ylabel("(meters)")
legend([L"x(t)", L"y(t)"])
title("Problem 3a: Extracted x, y data")

# part (b)

X = [ ... ] # for example, [t.^3 t.^0 3t.^2] makes a matrix where each row is t³ 1 3t²
xfit = X \ x  # least square solution

@show x₀ = xfit[1]
@show vx₀ = xfit[2]

plot(t, x, "bo") # data

tf = range(t[begin], t[end], length=1000) # finely spaced times for plotting curves
plot(tf, x₀ .+ vx₀ .* tf, "k-") # fit ... note that we use .+, .^, .* for elementwise operations
legend(["data", "fit"])
xlabel("time (seconds)")
ylabel("(meters)")
title("Problem 3b: Linear fit of x(t)")

# part (c)

Y = [ ... ] # FILL IN, SIMILAR TO part (b)
yfit = Y \ y  # least square solution

@show ... # SHOW YOUR FITTED y(0), vy(0), and g, similar to part (b)

plot(t, y, "r*") # data

... # PLOT THE PARABOLIC FIT and add labels, similar to part (b)

Problem 4 (10+5+5 points)

Suppose we have data (e.g. from some experimental measurement) $b_1,b_2,b_3,\ldots,b_{21}$ at the 21 equally spaced times $t = −10,−9,\ldots,9,10$. All measurements are $b_k = 0$ except that $b_{11} = 1$ at the middle time $t = 0$.

(a) Using least squares, what are the best $c$ and $d$ to fit those 21 points by a straight line $c+dt$?

(b) You are projecting the vector $b$ onto what subspace? (Give a basis.)

(c) Find a nonzero vector perpendicular to that subspace.

Problem 5 (10 points)

Suppose $\hat{x}$ is the least squares solution to $Ax \approx b$ (i.e. it minimizes $\Vert Ax-b\Vert$) and $\hat{y}$ is the least squares solution to $Ay \approx c$, where $A$ has full column rank. Does this tell you the least squares solution $\hat{z}$ to $Az \approx b + c$? If so, what is $\hat{z}$ and why?

Problem 6 (5+5+5 points)

(a) What matrix $P$ projects every vector in $\mathbb{R}^3$ onto the line that passes through origin and $a = [3, 4, 5]$ (column vector)?

(b) What is the nullspace of that matrix $P$? (Give a basis.)

(c) What is the row space of $P^7$?