In this worksheet, we will practice finding the general solution of a system of linear equations and determining a basis for its solution space.

**Q1: **

Find the general solution of the following system of linear equations: and then find a basis for its solution space.

- AThe general solution is . A basis for the solution space is .
- BThe general solution is . A basis for the solution space is .
- CThe general solution is . A basis for the solution space is .
- DThe general solution is . A basis for the solution space is .
- EThe general solution is . A basis for the solution space is .

**Q2: **

Find the general solution of the following system of linear equations: and then find a basis for its solution space.

- AThe general solution is . A basis for the solution space is .
- BThe general solution is . A basis for the solution space is .
- CThe general solution is . A basis for the solution space is .
- DThe general solution is . A basis for the solution space is .
- EThe general solution is . A basis for the solution space is .

**Q3: **

Find the general solution of the following system of linear equations: and then find a basis for its solution space.

- AThe general solution is . A basis for the solution space is .
- BThe general solution is . A basis for the solution space is .
- CThe general solution is . A basis for the solution space is .
- DThe general solution is . A basis for the solution space is .
- EThe general solution is . A basis for the solution space is .

**Q4: **

Find a basis for the solution space of the system

- A
- B
- C
- D
- E

**Q5: **

Find the general solution to the system of linear equations and hence find a basis for its solution space.

- AThe general solution is . A basis for the solution space is .
- BThe general solution is . A basis for the solution space is .
- CThe general solution is . A basis for the solution space is .
- DThe general solution is . A basis for the solution space is .
- EThe general solution is . A basis for the solution space is .

**Q6: **

Find the general solution of the system of linear equations and then find a basis for its solution space.

- AThe general solution is . A basis for the solution space is .
- BThe general solution is . A basis for the solution space is .
- CThe general solution is . A basis for the solution space is .
- DThe general solution is . A basis for the solution space is .
- EThe general solution is . A basis for the solution space is .

**Q7: **

Find the general solution of the system of linear equations and hence find a basis for its solution space.

- AThe general solution is , and a basis for the solution space is .
- B The general solution is , and a basis for the solution space is .
- CThe general solution is , and a basis for the solution space is .
- DThe general solution is , and a basis for the solution space is .
- E The general solution is , and a basis for the solution space is .

**Q8: **

Find the general solution of the system of linear equations and hence find a basis for its solution space.

- AThe general solution is , and a basis for the solution space is .
- B The general solution is , and a basis for the solution space is .
- CThe general solution is , and a basis for the solution space is .
- DThe general solution is , and a basis for the solution space is .
- EThe general solution is , and a basis for the solution space is .

**Q9: **

Let be the space of polynomials in the variable that have degree less than 4. Is a basis for this space?

- Ayes
- Bno

**Q10: **

Suppose that and are matrices and is a vector in the nullspace of . Consider this statement: Which of the following is true?

- AThe statement is true if and only if is a multiple of the identity matrix.
- BIf the statement is true, then is a multiple of the identity matrix.
- CThe statement is always true.
- DIf is a multiple of the identity matrix, then the statement is true.
- EIf is a multiple of the identity matrix, then the statement is false.

**Q11: **

Suppose that the columns of an matrix are linearly independent. Then, which of the following statements is always true?

- A has linearly independent eigenvectors.
- BThe determinant of is 1.
- C is idempotent.
- DThe kernal of is .
- E is a projection matrix.