The following problems from Elementary Linear Algebra, 6th Ed. are due Thursday, April 24.
To compute reduced row echelon form (RREF), feel free to use numerical applets such as this one on WolframAlpha, just cite which programs you've used, and write down the resulting matrix.
Section 5.3:
15
Section 5.6:
1(b,e), 5, 8
Section 6.1:
2(b,d), 4(b,d), 19
Section 6.2:
1(d,g)
Bonus 15 points out of 20. Section 6.3:
13
Hint: In part (c) of Exercise 6.3.13 (bonus), reduce to proving the following statement: If \( A \) is a \( 2\times 2 \) orthogonal matrix with \( \det(A)=1 \), then
$$
A = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}
$$
for some \( \theta\in\mathbb{R} \). This can be proved by writing the columns of \( A \) in polar coordinates, and using that they form an orthonormal set in \( \mathbb{R}^2 \), by orthogonality of \( A \).