Math 511: Linear Algebra |
1. Solve the linear system, if possible, using your favorite method.
\[
\begin{cases}
\phantom{2}x_1 + 2x_2 - \phantom{2}x_3 & = 1 \\
2x_1 - \phantom{2}x_2 + \phantom{2}x_3 & = 3 \\
-x_1 + 2x_2 + 3x_3 & = 7
\end{cases}
\]
2. The augmented matrix is in reduced row echelon form. Find the solution
of the corresponding system.
\[
\left(\begin{array}{rrrr|r}
1 & 5 & -2 & 0 & 3 \\
0 & 0 & 0 & 1 & 6 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0
\end{array}\right)
\]
3. Consider the matrices
\[
A = \left(\begin{array}{rrr}
3 & 5 & 1 \\
-2 & 0 & 2
\end{array}\right),\ \ \ \text{and}\ \ \
B = \left(\begin{array}{rr}
2 & 1 \\
1 & 3 \\
4 & 1
\end{array}\right).
\]
Compute \(AB\).
4. Given that \(R = \left(\begin{array}{rr}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta
\end{array}\right)\) for \(\theta \in \mathbb{R}\), show that \(R^{-1} = R^T\).
5. The matrix \(A = \left(\begin{array}{rr}
-2 & 4 \\
6 & 8
\end{array}\right)\) is nonsingular. Write \(A^{-1}\) as a product of elementary
matrices.
6. Find the inverse of the given matrix. Do not use determinants.
\[
A = \left(\begin{array}{rrr}
2 & 5 & 1\\
1 & 3 & 0 \\
2 & 5 & 0
\end{array}\right)
\]
7. Let \(A \in \mathbb{R}^{n\times n}\) be a nonsingular matrix. Compute
\[
\left(\begin{array}{c}
A^{-1} \\
I
\end{array}\right)
\left(\begin{array}{rr}
A & I
\end{array}\right).
\]
8. Find the \(LU\) Factorization of the matrix.
\[
A = \left(\begin{array}{r r r}
-2 & 1 & 2 \\
4 & 1 & -2 \\
-6 & -3 & 4
\end{array}\right)
\]
9. Consider the matrix:
\(
A = \left(\begin{array}{r r r}
2 & 3 & 5 \\
-2 & 4 & 2 \\
1 & -1 & 3
\end{array}\right)
\)
a.) Compute \(\det(A)\);
b.) Compute \(A^{-1}\), provided it exists. If it does not exist, explain
why.
10. Use Cramer's Rule to solve the system of equations. Give your answer
in simplified fraction form. If Cramer's Rule is not applicable, explain why.
\[
\begin{cases}
\phantom{-}5 x + 7 y & = \phantom{-}1 \\
-8 x + 3 y & = -1
\end{cases}
\]
11. Show that if \(Q \in \mathbb{R}^{n \times n}\) and \(Q^{-1} = Q^T\),
then either \(\det(Q) = 1\) or \(\det(Q) = -1\).
12. Find the values of \(\lambda\) that make \(A\) singular, where
\[
A = \left(\begin{array}{r r}
1 - \lambda & \sqrt{3} \\
\sqrt{3} & -1 - \lambda
\end{array}\right).
\]
13. Compute the determinant of \(A\), where
\[
A = \left(\begin{array}{r r r r}
1 & 0 & 0 & 0 \\
7 & 1 & 0 & 0 \\
\sqrt{2} & 0 & 1 & 0 \\
-3 & -\tfrac{1}{2} & 56 & 1
\end{array}\right)\ \left(\begin{array}{r r r r}
2 & 4 & -6 & 27 \\
0 & 1 & -9 & \sqrt{3} \\
0 & 0 & -2 & 115 \\
0 & 0 & 0 & 5
\end{array}\right).
\]
14. Consider the following vectors in \(\mathbb{R}^2\),
\[
\mathbf{u}_1 = \left(\begin{array}{r}
2 \\ 2
\end{array}\right),\ \ \ \mathbf{u}_2 = \left(\begin{array}{r}
3 \\ 1
\end{array}\right),\ \ \ \mathbf{v}_1 = \left(\begin{array}{r}
1 \\ -1
\end{array}\right),\ \ \ \text{and}\ \ \ \mathbf{v}_2 = \left(\begin{array}{r}
-2 \\ 3
\end{array}\right).
\]
Write \(\mathbf{x} = 4\mathbf{u}_1 - 2\mathbf{u}_2\) as a linear combination of
\(\mathbf{v}_1\) and \(\mathbf{v}_2\).
15. Consider the matrix,
\[
A = \left(\begin{array}{rrrr}
-3 & 1 & 3 & 4 \\
1 & 2 & -1 & -2 \\
6 & -3 & -5 & -7
\end{array}\right).
\]
Find bases for the row space, column space, and null space of \(A\). Clearly label
each answer.
16. Consider a matrix \(A \in \mathbb{R}^{3 \times 5}\) whose columns
\(\mathbf{a}_1,...,
\mathbf{a}_5\) satisfy \(\mathbf{a}_1\), \(\mathbf{a}_2\), and
\(\mathbf{a}_5\) are linearly independent, \(\mathbf{a}_3 =
\mathbf{a}_1 - \mathbf{a}_2\), and \(\mathbf{a}_4 = 2\mathbf{a}_1 + \mathbf{a}_3\).
a.) What is the reduced row echelon form (RREF) of \(A\)?
b.) What is the column space of \(A\)?
c.) what is the null space of \(A\)?
17. Let \(b \in \mathbb{R}\) be any real number, and define the operations
\(\oplus_b\) and \(\otimes_b\) on \(\mathbb{R}\) by:
\[
\begin{eqnarray*}
x \oplus_b y &=& x + y - b,\ \ \ \text{and} \\
\alpha\otimes_b x &=& \alpha x + (1 - \alpha)b
\end{eqnarray*}
\]
Is \((\mathbb{R},\oplus_b,\otimes_b)\) a vector space? Give sufficient proof of your answer.
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