# Worksheet: Null Space

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

Q1:

Find a basis for the solution space of the system

• A
• B
• C
• D
• E

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 .