ID: 1.2.35

Question: Suppose a coefficient matrix for a system has pivot columns. Is the system consistent? Why or why not?

Solution: The system is consistent if there is no row of the form $0 \ 0 \ 0 \ 0 \dots \ 1$ in the augmented matrix, which would indicate an inconsistency, as it would represent an impossible equation (e.g., $0 = 1$).

Correct answer:

• D. There is at least one row of the coefficient matrix that does not have a pivot position. This means the augmented matrix, which will have nine columns, must have a row of the form $0 \ 0 \ 0 \ 0 \ 0 \ 0 \ 0 \ 0 \ 1$, so the system is inconsistent.

ID: 1.3.17

Question: Let $a_1 = \begin{bmatrix} 1 \\ 4 \\ -1 \end{bmatrix}$, $a_2 = \begin{bmatrix} -7 \\ -23 \\ 2 \end{bmatrix}$, and $b = \begin{bmatrix} 3 \\ -3 \\ h \end{bmatrix}$. For what value(s) of $h$ is $b$ in the plane spanned by $a_1$ and $a_2$?

Solution: To find the value(s) of $h$ such that $b$ lies in the span of $a_1$ and $a_2$, we need to check if there exist scalars $x_1$ and $x_2$ such that: $x_1 a_1 + x_2 a_2 = b$ This leads to a system of equations. By solving it, we can determine the values of $h$.

• The value of $h$ is $h = 1$.

ID: 1.4.42

Question 1: Could a set of three vectors in $\mathbb{R}^4$ span all of $\mathbb{R}^4$? Explain.

Solution: To span $\mathbb{R}^4$, we need at least four vectors. A set of three vectors cannot span $\mathbb{R}^4$ because, in a matrix with three columns, we cannot have a pivot in each of the four rows.

Correct answer:

• D. No. The matrix $A$ whose columns are the three vectors has four rows. To have a pivot in each row, $A$ would have to have at least four columns (one for each pivot).

Question 2: Could a set of $n$ vectors in $\mathbb{R}^m$ span all of $\mathbb{R}^m$ when $n$ is less than $m$? Explain.

Solution: If $n < m$, there aren't enough vectors to span $\mathbb{R}^m$, because we need at least $m$ vectors to ensure that the matrix formed by the vectors has a pivot in every row.

Correct answer:

• D. No. The matrix $A$ whose columns are the $n$ vectors has $m$ rows. To have a pivot in each row, $A$ would have to have at least $m$ columns (one for each pivot).

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Additional Related Questions:

1. What is the definition of a "pivot position" in the context of a matrix?
2. How can the consistency of a system be checked using the row echelon form of a matrix?
3. What is the geometric interpretation of the span of vectors?
4. How does the rank of a matrix relate to the number of pivot positions?
5. How can you determine if a vector lies in the span of other vectors?

Tip: When solving systems of equations, check for any rows that lead to contradictions, such as $0 = 1$, which would make the system inconsistent.