Review Problems for Exam 1

Select all pairs of parallel vectors.
\(\begin{bmatrix}-3\\-6\\3\end{bmatrix}\), \(\begin{bmatrix}5\\10\\-5\end{bmatrix}\) \(\begin{bmatrix}3\\-6\\3\end{bmatrix}\), \(\begin{bmatrix}-1\\2\\-1\end{bmatrix}\) \(\begin{bmatrix}1\\0\\1\end{bmatrix}\), \(\begin{bmatrix}-1\\0\\1\end{bmatrix}\) \(\begin{bmatrix}2\\0\\-1\end{bmatrix}\), \(\begin{bmatrix}-8\\0\\4\end{bmatrix}\)
Find \(Q\) such that \(\overrightarrow {PQ}=\vec {v}\) if
\[P(-1, 2, 2), \vec {v}=\begin{bmatrix}1\\3\\1\end{bmatrix}\]
\[Q(\answer {0},\answer {5},\answer {3})\]
Find a unit vector \(\vec {u}\) in the direction of \(\begin{bmatrix}7\\-1\\5\end{bmatrix}\)
\[\vec {u}=\answer {\frac {1}{\sqrt {75}}}\begin{bmatrix}7\\-1\\5\end{bmatrix}\]
Enter the exact answer. No decimal approximations.
Select all pairs of orthogonal vectors.
\(\begin{bmatrix}1\\-3\end{bmatrix}\), \(\begin{bmatrix}-3\\1\end{bmatrix}\) \(\begin{bmatrix}2\\3\end{bmatrix}\), \(\begin{bmatrix}-6\\4\end{bmatrix}\) \(\begin{bmatrix}1\\2\\-1\end{bmatrix}\), \(\begin{bmatrix}4\\-2\\0\end{bmatrix}\) \(\begin{bmatrix}-1\\3\\-2\\5\end{bmatrix}\), \(\begin{bmatrix}2\\1\\-2\\-1\end{bmatrix}\)
Use the augmented matrix notation and elementary row operations to find all solutions (if any) to the system.
\[\begin{matrix} 3x& -&y&=&0\\ 2x & -&3y&= &1 \end{matrix}\]
\[(\answer {-1/7}, \answer {-3/7})\]
Enter your answers using fractions. If the system is inconsistent, type DNE in each answer cell.
Use the augmented matrix notation and elementary row operations to find all solutions (if any) to the system.
\[\begin{matrix} 3x& -&y&=&2\\ -6x & +&2y&= &-4 \end{matrix}\]
\[(\answer {(1/3)t+2/3}, t)\]
Enter your answers using fractions.
Use the augmented matrix notation and elementary row operations to find all solutions (if any) to the system.
\[\begin{matrix} -2x& +&3y&+&3z&=&-9\\ 3x & -&4y&+&z&= &5\\ -5x&+&7y&+&2z&=&-14 \end{matrix}\]
\[(\answer {-15t-21}, \answer {-11t-17}, t)\]
If the system is inconsistent, type DNE in each answer cell.
Use the augmented matrix notation and elementary row operations to find all solutions (if any) to the system.
\[\begin{matrix} x& +&2y&-&z&=&2\\ 2x & +&5y&-&3z&= &1\\ x&+&4y&-&3z&=&3 \end{matrix}\]
\[(\answer {DNE}, \answer {DNE}, \answer {DNE})\]
If the system is inconsistent, type DNE in each answer cell.
Use the augmented matrix notation and elementary row operations to find all solutions (if any) to the system.
\[\begin{matrix} 3x& -&2y&+&z&=&-2\\ x & -&y&+&3z&= &5\\ -x&+&y&+&z&=&-1 \end{matrix}\]
\[(\answer {-7}, \answer {-9}, \answer {1})\]
If the system is inconsistent, type DNE in each answer cell.
The augmented matrix of a system of linear equations has been carried to the following by row operations. Solve the system.
\[\left [\begin{array}{cccccc|c} 1 & -2 & 0 & 2 & 0 & 1 & 1\\ 0 & 0 & 1 & 5 & 0 & -3 & -1\\ 0 & 0 & 0 & 0 & 1 & 6 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right ]\]
\[(\answer {1-t-2s+2r}, r, \answer {-1+3t-5s}, s,\answer {1-6t}, t)\]
The augmented matrix of a system of linear equations has been carried to the following by row operations. Solve the system.
\[\left [\begin{array}{ccccc|c} 1 & -1 & 2 & 4 & 6 & 2\\ 0 & 1 & 2 & 1 & -1 & -1\\ 0 & 0 & 0 & 1 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right ]\]
Is this matrix in reduced row-echelon form? If not, carry it to the reduced row-echelon form.
\[(\answer {-4s-5t-4}, \answer {-2+t-2s}, s,\answer {1}, t)\]
Consider the following augmented matrix in which \(\ast \) denotes an arbitrary number and \(\blacksquare \) denotes a nonzero number.
\begin{equation*} \left [ \begin{array}{ccccc|c} \blacksquare & \ast & \ast & \ast & \ast & \ast \\ 0 & \blacksquare & \ast & \ast & \ast & \ast \\ 0 & 0 & \blacksquare & \blacksquare & \ast & \ast \\ 0 & 0 & 0 & 0 & \blacksquare & 0 \end{array} \right ] \end{equation*}

Which of the following is true? Select all that apply.

The corresponding system of equations is consistent. The corresponding system of equations is inconsistent. The rank of the augmented matrix is \(4\). The rank of the augmented matrix is \(5\). The rank of the augmented matrix is \(6\). There are no free variables. There is one free variable. There are two free variables. The corresponding system has a unique solution. The corresponding system has infinitely many solutions.
Consider an augmented matrix of the form
\[\left [\begin{array}{c|c} A & \vec {b}\\ \end{array}\right ]\]
corresponding to a system of linear equations.

True or False?

1.
If there is more than one solution, \(\left [\begin{array}{c|c} A & \vec {b}\\ \end{array}\right ]\) has a row of zeros.
True False
2.
If \(\left [\begin{array}{c|c} A & \vec {b}\\ \end{array}\right ]\) has a row of zeros, there is more than one solution.
True False
3.
If there is no solution, \(\mbox {rref}(A)\) has a row of zeros.
True False
4.
If \(\mbox {rref}(A)\) has a row of zeros, there is no solution.
True False
5.
There is NO system that is inconsistent for every \(\vec {b}\).
True False
6.
If the system is consistent for some choice of \(\vec {b}\), it is consistent for every choice of \(\vec {b}\).
True False
Express \(\begin{bmatrix}127\\-26\end{bmatrix}\) as a linear combination of \(\begin{bmatrix}101\\50\end{bmatrix}\) and \(\begin{bmatrix}25\\42\end{bmatrix}\).
\[\answer {2}\begin{bmatrix}101\\50\end{bmatrix}+\answer {-3}\begin{bmatrix}25\\42\end{bmatrix}=\begin{bmatrix}127\\-26\end{bmatrix}\]
Suppose \(\vec {v}_1, \vec {v}_2, \vec {v}_3\) and \(\vec {w}\) are vectors in \(\RR ^n\). What are the implications of the statement: \(\vec {w}\) is in the span of \(\vec {v}_1, \vec {v}_2, \vec {v}_3\) Select all that apply.
Vector \(\vec {w}\) a linear combination of \(\vec {v}_1, \vec {v}_2, \vec {v}_3\) The set \(\{\vec {v}_1, \vec {v}_2, \vec {v}_3, \vec {w}\}\) contains redundant vectors. Vectors \(\vec {v}_1, \vec {v}_2, \vec {v}_3, \vec {w}\) are linearly independent. Vectors \(\vec {v}_1, \vec {v}_2, \vec {v}_3, \vec {w}\) are linearly dependent. Vector \(\vec {v}_1\) can be written as a linear combination of \(\vec {v}_2, \vec {v}_3, \vec {w}\). The equation \(a\vec {v}_1+ b\vec {v}_2+ c\vec {v}_3+ d\vec {w}=\vec {0}\) has a non-trivial solution.
Express \(\begin{bmatrix}-1\\4\\5\end{bmatrix}\) as a linear combination of \(\begin{bmatrix}2\\1\\-3\end{bmatrix}\), \(\begin{bmatrix}-1\\1\\4\end{bmatrix}\) and \(\begin{bmatrix}1\\1\\-2\end{bmatrix}\).

If this is not possible, enter DNE in each answer cell.

\[\begin{bmatrix}-1\\4\\5\end{bmatrix}=\answer {-1}\begin{bmatrix}2\\1\\-3\end{bmatrix}+\answer {2}\begin{bmatrix}-1\\1\\4\end{bmatrix}+\answer {3}\begin{bmatrix}1\\1\\-2\end{bmatrix}\]
Suppose three planes given by the equations below intersect at a single point.
\[\begin{matrix} a_1x& +&a_2y&+&a_3z&=&0\\ b_1x & +&b_2y&+&b_3z&= &0\\ c_1x&+&c_2y&+&c_3z&=&0 \end{matrix}\]

What can be said about the vectors \(\begin{bmatrix}a_1\\b_1\\c_1\end{bmatrix}\), \(\begin{bmatrix}a_2\\b_2\\c_2\end{bmatrix}\) and \(\begin{bmatrix}a_3\\b_3\\c_3\end{bmatrix}\)?

Select all that apply.

Nothing can be deduced from the information given. The three vectors are linearly independent. The three vectors are linearly dependent. The three vectors are pairwise orthogonal. The first vector can be written as a linear combination of the other two. The equation \(\alpha \begin{bmatrix}a_1\\b_1\\c_1\end{bmatrix}+\beta \begin{bmatrix}a_2\\b_2\\c_2\end{bmatrix}+\omega \begin{bmatrix}a_3\\b_3\\c_3\end{bmatrix}=\vec {0}\) has only the trivial solution.
Use the navigation bar arrows in the interactive below to advance the construction steps. Use the final result to express \(\vec {w}\) as a linear combination of \(\vec {u}\) and \(\vec {v}\).

\[\vec {w}=\answer {-1}\vec {u}+\answer {0.5}\vec {v}\]
True or False?
1.
If \(\{\vec {x}, \vec {y}\}\) is linearly independent, then \(\{\vec {x}, \vec {y}, \vec {x}+\vec {y}\}\) is linearly independent.
True False
2.
If \(\{\vec {x}, \vec {y}, \vec {z}\}\) is linearly independent, then \(\{\vec {y}, \vec {z}\}\) is linearly independent.
True False
3.
If \(\{\vec {y}, \vec {z}\}\) is linearly dependent, then \(\{\vec {x}, \vec {y}, \vec {z}\}\) is linearly dependent.
True False
4.
If \(a\vec {x}+b\vec {y}+c\vec {z}=\vec {0}\), then \(\{\vec {x}, \vec {y}, \vec {z}\}\) is linearly independent.
True False
Find the inverse of the following matrix.
\[A=\begin{bmatrix}1 & -1 & 2\\-5 & 7 & -11\\-2 & 3 &-5\end{bmatrix}\]
\[A^{-1}=\begin{bmatrix}\answer {2} & \answer {-1} & \answer {3}\\\answer {3} & \answer {1} & \answer {-1}\\\answer {1} & \answer {1} &\answer {-2}\end{bmatrix}\]
Consider the system of equations:
\[\begin{matrix} x& -&3y&+&z&=&2\\ -3x & -&4y&+&z&= &0\\ & &2y&-&z&=&1 \end{matrix}\]

Let \(A\) be the coefficient matrix corresponding to this system, and let \(\vec {b}=\begin{bmatrix}2\\0\\1\end{bmatrix}\).

Suppose that we find that \((1, -2, -5)\) is a unique solution to this system. What does this tell us? Select ALL that apply.

\(A\) is invertable. Vector \(\vec {b}\) is in the span of the columns of \(A\). Columns of \(A\) are linearly dependent. Equation \(A\vec {x}=\vec {b}\) has a unique solution. Vector \(\vec {b}\) can be written as a linear combination of the columns of \(A\). \(A\vec {b}=\begin{bmatrix}1\\-2\\-5\end{bmatrix}\)
Let \(A=\begin{bmatrix}|&|&|\\\vec {c}_1& \vec {c}_2 & \vec {c}_3\\|&|&|\end{bmatrix}\) be a \(3\times 3\) matrix with columns \(\vec {c}_1\), \(\vec {c}_2\) and \(\vec {c}_3\).

Suppose \(A\begin{bmatrix}-1\\-1\\2\end{bmatrix}=\begin{bmatrix}0\\1\\1\end{bmatrix}\)

Which of the following can we conclude from the given information? Select ALL that apply.

\(\vec {c}_2+\vec {c}_3=\begin{bmatrix}-1\\-1\\2\end{bmatrix}\) \(-\vec {c}_1-\vec {c}_2+2\vec {c}_3=\begin{bmatrix}0\\1\\1\end{bmatrix}\) \(A\) is non-singular. \(\begin{bmatrix}0\\1\\1\end{bmatrix}\) is in \(\text {span}(\vec {c}_1, \vec {c}_2, \vec {c}_3)\). Vector \(\begin{bmatrix}0\\1\\1\end{bmatrix}\) can be written as a linear combination of the columns of \(A\).
Suppose \(A\) is a non-singular \(3\times 3\) matrix. Let \(A^{-1}=\begin{bmatrix}|&|&|\\\vec {d}_1& \vec {d}_2 & \vec {d}_3\\|&|&|\end{bmatrix}\).

Suppose \(A\begin{bmatrix}2\\3\\4\end{bmatrix}=\begin{bmatrix}5\\6\\7\end{bmatrix}\)

Express \(\begin{bmatrix}2\\3\\4\end{bmatrix}\) as a linear combination of the columns of \(A^{-1}\).

\[\begin{bmatrix}2\\3\\4\end{bmatrix}=\answer {5}\vec {d}_1+\answer {6}\vec {d}_2+\answer {7}\vec {d}_3\]
Let \(A\) be a \(2\times 5\) matrix. Compute the dimensions of each product.

\(A^TA\) is a \(\answer {5}\times \answer {5}\) matrix.

\(AA^T\) is a \(\answer {2}\times \answer {2}\) matrix.

Let \(A=\begin{bmatrix}1 & 2\\-1 & 1\end{bmatrix}\) and \(C=\begin{bmatrix}-1 & 1\\2 & 1\end{bmatrix}\). Find elementary matrices \(E_1\) and \(E_2\) such that \(C=E_2E_1A\).
\[E_1=\begin{bmatrix}\answer {1} & \answer {-1}\\\answer {0} & \answer {1}\end{bmatrix};\quad E_2=\begin{bmatrix}\answer {0} & \answer {1}\\\answer {1} & \answer {0}\end{bmatrix}\]