# Worksheet: Null Space

In this worksheet, we will practice defining the null space of a matrix and finding its basis and dimension.

**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: **

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 is a multiple of the identity matrix, then the statement is false.
- CThe statement is always true.
- DIf is a multiple of the identity matrix, then the statement is true.
- EIf the statement is true, then is a multiple of the identity matrix.

**Q4: **

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 .
- BThe 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 .

**Q5: **

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 .
- BThe 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 .

**Q6: **

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 .

**Q7: **

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 .

**Q8: **

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 .

**Q9: **

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 .