Linear Algebra with Applications, 10th Edition

Section 3.2 Exercises

1. Determine whether the following sets form subspaces of ${ℝ}^2$:

   1. $\{(x_1,x_2{)}^T|x_1+x_2=0\}$

   2. $\{(x_1,x_2{)}^T|x_1{x}_2=0\}$

   3. $\{(x_1,x_2{)}^T|x_1=3x_2\}$

   4. $\{(x_1,x_2{)}^T|\left|x_1\right|=\left|x_2\right|\}$

   5. $\{(x_1,x_2{)}^T|x_1^2=x_2^2\}$

2. Determine whether the following sets form subspaces of ${ℝ}^3$:

   1. $\{(x_1,x_2,x_3{)}^T|x_1+x_3=1\}$

   2. $\{(x_1,x_2,x_3{)}^T|x_1=x_2=x_3\}$

   3. $\{(x_1,x_2,x_3{)}^T|x_3=x_1+x_2\}$

   4. $\{(x_1,x_2,x_3{)}^T|x_3=x_1\text{ or }{x}_3=x_2\}$

3. Determine whether the following are subspaces of ${ℝ}^{2\times 2}$:

   1. The set of all $2\times 2$ diagonal matrices

   2. The set of all $2\times 2$ triangular matrices

   3. The set of all $2\times 2$ lower triangular matrices

   4. The set of all $2\times 2$ matrices A such that $a_{12}=1$

   5. The set of all $2\times 2$ matrices B such that $b_{11}=0$

   6. The set of all symmetric $2\times 2$ matrices

   7. The set of all singular $2\times 2$ matrices

4. Determine the null space of each of the following matrices:

   1. $[\begin{array}{cc}2& 1\end{array}$