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Worksheet: Basis for the Kernel of a Matrix

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 a basis for the solution space of the following system.

A
B
C
D
E

Q2:

Find the general solution of the following 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 .

Q3:

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 .

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 .
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 .

Q5:

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

Ayes
Bno

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 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 .

Q8:

Find the general solution of the following 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 .

Q9:

Find the general solution of the following 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 .