MATH 345: Linear Algebra and Optimization

Activity 3.1.2. Vector length as a scalar-valued function (U3-LO1).

Activity 3.1.3. Component functions of a matrix map (U3-LO1).

Activity 3.1.4. Corners are not enough (U3-LO1).

1. Compute the images of the four corners of the unit square under \(T\text{.}\)

   🔗
   🔗
2. Compute the images of the four corners of the unit square under \(F\text{.}\)

   🔗
   🔗
3. Compute the images of \(\begin{bmatrix}0\\1/2\end{bmatrix}\) under both maps.

   🔗
   🔗
4. What does \(F\) do to a point \(\begin{bmatrix}0\\y\end{bmatrix}\) on the left edge of the unit square?

   🔗
   🔗

Activity 3.1.6. Reading a square-grid computation (U3-LO1).

import numpy as np

def F(P):
    x = P[0, :]
    y = P[1, :]
    return np.vstack([x + y**2, y])

P = np.array([
    [0.0, 1.0, 1.0, 0.0, 0.0],
    [0.0, 0.0, 1.0, 1.0, 0.5],
])

F(P)

array([[0.  , 1.  , 2.  , 1.  , 0.25],
       [0.  , 0.  , 1.  , 1.  , 0.5 ]])

1. What points are stored in the first four columns of \(P\text{?}\)

   🔗
   🔗
2. What point is stored in the last column of \(P\text{?}\)

   🔗
   🔗
3. What output corresponds to the last column?

   🔗
   🔗
4. Why do the first four columns not detect the difference between \(F\) and the shear \(T(x,y)=(x+y,y)\text{?}\)

   🔗
   🔗
5. Why is a grid more informative than the four corners?

   🔗
   🔗

Activity 3.1.10. Sketching domains (U3-LO1).

(a)

\(f(x,y) = \ln(y - x^2) + \sqrt{x} \text{.}\)

🔗

Solution.

The expression that defines the function \(f\) consists of the sum of two terms, and so to find the domain of \(f\text{,}\) we must identify where both terms are well defined.

🔗

The term \(\ln(y - x^2)\) is well defined when the input to \(\ln\) is positive, i.e., when \(y - x^2 > 0\text{.}\) We may rearrange this inequality to read \(y > x^2\)

🔗

The term \(\sqrt{x}\) is well-defined when the input to \(\sqrt{\cdot}\) is non-negative, i.e., when \(x \geq 0\text{.}\)

🔗

So \(f\) is well-defined when both \(y > x^2\) and when \(x \geq 0\text{,}\) and so the domain of \(f\) is the set of all points \((x,y)\) such that \(x \geq 0\) and \(y > x^2\text{.}\) Using set notation, we can also write the domain as the set

\begin{equation*} \{ (x,y) \in \R^2: x \geq 0\ \text{and}\ y > x^2 \}\text{.} \end{equation*}

The set is the region in the plane bounded from below by the parabola \(y = x^2\text{,}\) and on the left by the \(y\)-axis, as depicted in Figure 3.1.11.

🔗



The figure shows coordinate axes with a vertical boundary at \(x=0\) and an upward-opening parabola \(y=x^2\text{.}\) The shaded purple region lies to the right of the \(y\)-axis and above the parabola. The drawing emphasizes that the domain requires both \(x\geq 0\) and \(y>x^2\text{.}\)

🔗

🔗

🔗

(c)

\(f(x,y) = \frac{\sqrt{x}}{\sqrt{y}}\)

🔗

Solution.

The expression defining \(f\) is a fraction with numerator \(\sqrt{x}\) and denominator \(\sqrt{y}\text{.}\) The expression is therefore well-defined when both the numerator and denominator are well defined, and when the denominator is non-zero.

🔗

The numerator \(\sqrt{x}\) is well-defined when \(x \geq 0\text{.}\) The denominator \(\sqrt{y}\) is well-defined when \(y \geq 0\text{,}\) and is non-zero when \(y > 0\text{.}\)

🔗

So the domain of \(f\) is the set of all \((x,y) \in \R^2\) with \(x \geq 0\) and \(y > 0\text{,}\) or in set notation, the set

\begin{equation*} \{(x,y) \in \mathbb{R}^2 : x \geq 0\ \text{ and }\ y > 0\}\text{.} \end{equation*}

This set is the first quadrant in the \(xy\)-plane, including the positive \(y\)-axis, but excluding the origin and the positive \(x\)-axis. See Figure 3.1.12.

🔗



The coordinate plane is shaded purple in the region with \(x\geq 0\) and \(y>0\text{.}\) The \(y\)-axis is included as part of the boundary, while the \(x\)-axis is drawn as a dashed red boundary to show it is excluded. A handwritten note points out that the dashed boundary is where \(y=0\text{.}\)

🔗

🔗

🔗

Activity 3.1.16. Sketching level curves (U3-LO1).

Solution.



The left side lists level values such as \(1\text{,}\) \(e\text{,}\) \(e^4\text{,}\) and \(e^9\) with the corresponding equations for \(x^2+y^2\text{.}\) The right side shows concentric circles centered at the origin in the \(xy\)-plane. Larger level values correspond to circles farther from the origin.

🔗

🔗