Linear Algebra with Applications, 10th Edition

Chapter Test B

1. 1. Hint: Find a nontrivial linear combination of ${\mathbf{\text{x}}}_1,{\mathbf{\text{x}}}_2,{\mathbf{\text{x}}}_3$ that equals 0.

2. 2. 

   1. (b) Hint: $S_2$ consists of all vectors of the form

      $$\left[\begin{array}{c}\mathit{\text{a}}\\ \mathbf{0}\end{array}\right]\mathrm{or}\left[\begin{array}{c}\mathbf{0}\\ \mathit{\text{b}}\end{array}\right]$$

3. 5. Hint: One-dimensional subspaces correspond to lines through the origin, and two-dimensional subspaces correspond to planes through the origin.

4. 6. Hint: The vectors in S are matrices of the form

   $$\left[\begin{array}{cc}a& b\\ b& c\end{array}\right]$$

5. 7. 

   1. (c) Hint: The consistency theorem guarantees that there will be at least one solution. Whether or not there are more depends on whether there are free variables in the echelon form of the coefficient matrix.

6. 9. Hint: You should be able to show linear independence by just working from the definition.

7. 10. 

   1. (a) Hint: First determine the rank of A.

   2. (b) Hint: If A and U are row equivalent ...