Linear Algebra for Machine Learning and Data Science (3)

Systems of linear equations

Quiz 1

Q1.

A scientist has 6 weights, three of type X, and three of type Y. She wants to determine the weight of each type since the labels have worn off, but she has a problem. Her scale lacks sufficient precision for objects weighing less than 1000 grams.
Initially, the scientist placed all three X weights on the scale and received the message "Insufficient weight for an accurate measurement." The same happened with the three Y weights. Therefore, the scientist decided to perform two weighings:

1. She weighed the 3 X weights together with 1 Y weight and obtained a total weight of 1100 grams.
2. She weighed the 3 Y weights together with 1 X weight and obtained a total weight of 1050 grams.

Assuming X weight weighs $x$ grams and Y weight weighs $y$ grams, the best linear system that describes the experiment is:

Choice

1

$$ \begin{cases} 3x + y = 1050 \\ x + 3y = 1100 \end{cases} $$

2

$$ \begin{cases} 3x + y = 1100 \\ x + 3y = 1050 \end{cases} $$

3

$$ \begin{cases} 3x + 3y = 1100 \\ 3x + 3y = 1050 \end{cases} $$

4

$$ \begin{cases} 3x = 1100 \\ 3y = 1050 \end{cases} $$

Answer

2

You've successfully translated the story into a linear system!

Q2.

Which of the following matrices can be used to determine the singularity of the system of equations below?

$$ \begin{cases} 2x + 3y = 15\\ 2x + 4y = 16 \end{cases} $$

Choice

1

$$ \left[\begin{array}{cc}2 & 15 \\2 & 16\end{array}\right] $$

2

$$ \left[\begin{array}{cc}3 & 15 \\4 & 16\end{array}\right] $$

3

$$ \left[\begin{array}{cc}2 & 3 \\2 & 4\end{array}\right] $$

4

$$ \left[\begin{array}{cc}2 & 2 \\3 & 4\end{array}\right] $$

Answer

3

This is the correct representation of the system of equations in matrix form. The coefficients in the first row are the coefficients of x and y in the first equation and the same logic applies to the second row and second equation.

Q3.

Consider the next three plots below.

Now, consider the next three systems of equations below.

System 1

$\begin{cases} 3x - 2y = 1\\ x + y = 3 \end{cases}$

System 2

$\begin{cases} 3x + 3y = 2\\ 9x + 9y = 6 \end{cases}$

System 3

$\begin{cases} x +3y = 4\\x + 3y = 3 \end{cases}$

Each plot represents one of the systems described. Choose the correct option.

Choice

1

Plot 1 represents System 3

Plot 2 represents System 1

Plot 3 represents System 2

2

Plot 1 represents System 2

Plot 2 represents System 1

Plot 3 represents System 3

3

Plot 1 represents System 1

Plot 2 represents System 2

Plot 3 represents System 3

4

Plot 1 represents System 1

Plot 2 represents System 3

Plot 3 represents System 2

5

Plot 1 represents System 3

Plot 2 represents System 2

Plot 3 represents System 1

Answer

1

Well done! Plot 1 represents two parallel and distinct lines, so it must represent a contradictory system, which is System 3 since it assigns two distinct values (4 and 3) to the same quantity $(x+3y)$.

Plot 2 represents two lines that intersect at one point, so it must represent a system with a single solution. Note that System 2 has the second line as two times the first one $(9x+9y=6$ is equivalent to $3⋅(3x+3y)=3⋅2)$, so it can't be System 2, remaining only System 1. It can also be noted that the equations in System 1 are not a multiple of each other.

Finally, Plot 3 represents two identical lines, so it must represent a redundant system. As discussed above, it is System 2.

Q4.

Calculate the determinant of the following matrix. Is the matrix singular or non-singular?

Hint: To find the determinant apply the formula ad−bc. A matrix of determinant 0 is singular, while a determinant different than 0 represents a complete system, thus a non-singular matrix.

Choice

1. det(A) = −2. The matrix is singular.

2. det(A) = 2. The matrix is non-singular.

3. det(A) = 0. The matrix is singular.

4. det(A) = 2. The matrix is singular.

Answer

2

You have correctly calculated the determinant and identified the singularity of the matrix.

Q5.

Determine if this matrix has linearly dependent or independent rows.

Choice

1. Linearly independent

2. Linearly dependent

3. It cannot be determined

Answer

1

The matrix has linearly independent rows. You cannot obtain one row by using row operations on the other row.

All the information here is based on the Linear Algebra for Machine Learning and Data Science | Coursera from DeepLearning.AI