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Quiz 1: Linear Algebra and Differential Equations (Summer 2022)

Question 1

You must take this quiz completely alone. Showing it to or discussing it with anyone else is forbidden. Reproducing or sharing this material on any other websites or platforms is also forbidden. You are permitted to consult your notes, the textbook, any materials handed out in class, and any materials on the bCourses site. You may not consult any other resources (for instance public websites, Wolfram Alpha, and calculators).

I have read and understood these instructions. I agree that I will follow both the instructions and the Berkeley Honor Code.

I will not follow these instructions or the Berkeley Honor Code.

Question 2

True or false: If the first column of an augmented matrix is not a pivot column then the associated linear system must be inconsistent.

True

False

Consider an augmented matrix (of any size) consisting of all zeros. The associated linear system is consistent. In fact its solution set is everything! A system is only inconsistent if the last column of the augmented matrix is a pivot column.

Question 3

True or false: Consider the matrix

The system       has a unique solution for any .

True

False

We notice that the matrix is already in REF form, and there is not pivot in the third column. This means that the third variable,, is free, so there will be infinitely many solutions to for each .

Notice that is always consistent since has a pivot in each row.

Question 4

Select all of the equations that are linear in the variables ,, and.

Question 5

Alice has the following linear system of equations, and wants to know how many solutions it has:

How many solutions does the linear system have?

Zero

Exactly one

Infinitely many

Recall that a system has a solution if and only if there is no pivot in the last column of its augmented matrix. Also, recall that solutions are unique if and only if every column of the coefficient matrix has a pivot.

To make these decisions, we usually calculate the row echelon form of the augmented matrix:

Here, the augmented matrix is already in row echelon form. Each of the first four columns has a pivot, so there is exactly one solution.

Select all of the sets of vectors that span .

ou Answered

orrect Answer

We know that a set of vectors in span if and only if the matrix with those vectors as its columns has a pivot in each row.

This makes it relatively easy to check each of these answers. The set that has only vectors cannot span , since the matrix will only have columns and therefore at most pivots.

This means there can't be a pivot in each row.

The set that includes the zero vector cannot span for a similar reason. The matrix will have a column of all zeros, which won't have a pivot, so there are at most pivots, meaning there isn't a pivot in each row.

For the other two sets of vectors, after constructing the matrix, you should see relatively quickly that each row does indeed have a pivot!

Question 7

Consider the following linear system:

If the system is consistent, select the value of for any solution.

Otherwise, select that the system is inconsistent.

-9

-11

The system is inconsistent.

-10

Using matrix techniques, we row reduce the augmented matrix to reduced row echelon form:

From this, we can read off (1) that the system is consistent since there is no pivot in the last column and (2) that is the - entry of the RREF.

Question 8

Determine the number of pivots in the following matrix.

ou Answered

To determine the number of pivots, we row reduce:

The number of pivots in is the number of leading 's in the reduced row echelon form of , which is .

A couple of observations: (1) there is at least one pivot because is nonzero, (2) there are at most three pivots because there are only three rows.

Question 9

Consider the list of vectors

For what value of is the third vector a linear combination of the first two?

2

-1

3

0

The question amounts to when the linear system is consistent.

Row reducing the augmented matrix for the system as far as we can, we get

The system is consistent when there is no pivot in the last column, which is when .

Question 10

Find scalars , and such that

Report as your answer.

4

4 (with margin: 0)

We can rewrite the given vector equation as a matrix equation:

We solve this to find :

This gives us the results that .

If you want to follow along, we applied the following row operations:

Question 11

Consider the matrix equation

For what values of does this equation have solutions?

0 (with margin: 0)

A linear system has a solution if and only if the last column of the augmented matrix does not have a pivot. To calculate this, we row reduce the augmented matrix as far as we can:

If you want to follow along, we applied the following row operations:

Thus, there is no pivot in the last column if , and so we

solve this to get the only value for which the system is consistent.