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

  • AThe general solution is 4 𝑡 2 𝑡 3 𝑡 . A basis for the solution space is 4 2 3 .
  • BThe general solution is 𝑡 𝑡 3 𝑡 . A basis for the solution space is 1 1 3 .
  • CThe general solution is 3 𝑡 2 𝑡 4 𝑡 . A basis for the solution space is 3 2 4 .
  • DThe general solution is 3 𝑡 𝑡 𝑡 . A basis for the solution space is 3 1 1 .
  • EThe general solution is 3 𝑡 𝑡 𝑡 . A basis for the solution space is 1 1 3 .

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 𝑡 𝑡 3 𝑡 . A basis for the solution space is 1 1 3 .
  • BThe general solution is 𝑡 2 𝑡 4 𝑡 . A basis for the solution space is 1 2 4 .
  • CThe general solution is 3 𝑡 𝑡 𝑡 . A basis for the solution space is 3 1 1 .
  • DThe general solution is 4 𝑡 2 𝑡 𝑡 . A basis for the solution space is 4 2 1 .
  • EThe general solution is 4 𝑡 2 𝑡 𝑡 . A basis for the solution space is 1 2 4 .

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 𝑡 2 𝑡 4 𝑡 . A basis for the solution space is 1 2 4 .
  • BThe general solution is 𝑡 2 𝑡 3 𝑡 . A basis for the solution space is 1 2 3 .
  • CThe general solution is 4 𝑡 2 𝑡 𝑡 . A basis for the solution space is 4 2 1 .
  • DThe general solution is 3 𝑡 2 𝑡 𝑡 . A basis for the solution space is 3 2 1 .
  • EThe general solution is 3 𝑡 2 𝑡 𝑡 . A basis for the solution space is 1 2 3 .

Q4:

Find a basis for the solution space of the system

  • A 2 1 5
  • B 1 2 1
  • C 5 1 2
  • D 1 2 1
  • E 1 2 1

Q5:

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

  • AThe general solution is 𝑡 2 𝑡 𝑡 0 . A basis for the solution space is 1 2 1 0 .
  • BThe general solution is 0 𝑡 𝑡 𝑡 . A basis for the solution space is 0 1 1 1 .
  • CThe general solution is 0 𝑡 2 𝑡 𝑡 . A basis for the solution space is 0 1 2 1 .
  • DThe general solution is 𝑡 𝑡 𝑡 0 . A basis for the solution space is 1 1 1 0 .
  • EThe general solution is 𝑡 𝑡 𝑡 0 . A basis for the solution space is 0 1 1 1 .

Q6:

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

  • AThe general solution is 𝑡 𝑡 3 𝑡 3 𝑡 . A basis for the solution space is 1 1 3 3 .
  • BThe general solution is 𝑡 𝑡 𝑡 0 . A basis for the solution space is 1 1 1 0 .
  • CThe general solution is 𝑡 𝑡 3 𝑡 3 𝑡 . A basis for the solution space is 1 1 3 3 .
  • DThe general solution is 0 𝑡 𝑡 𝑡 . A basis for the solution space is 0 1 1 1 .
  • EThe general solution is 0 𝑡 𝑡 𝑡 . A basis for the solution space is 1 1 1 0 .

Q7:

Find the general solution of the system of linear equations 1 1 0 1 1 1 1 0 3 1 1 2 3 3 0 3 𝑥 𝑦 𝑧 𝑤 = 0 0 0 0 , and hence find a basis for its solution space.

  • AThe general solution is 𝑠 𝑡 𝑠 𝑡 𝑠 𝑡 , and a basis for the solution space is 1 1 1 0 , 1 1 0 1 .
  • B The general solution is 𝑡 1 2 𝑠 1 2 𝑡 𝑠 1 2 𝑠 1 2 𝑡 , and a basis for the solution space is 0 1 2 1 1 2 , 1 1 2 0 1 2 .
  • CThe general solution is 𝑡 𝑠 𝑡 𝑠 𝑠 𝑡 , and a basis for the solution space is 0 1 1 1 , 1 1 0 1 .
  • DThe general solution is 1 2 𝑠 1 2 𝑡 1 2 𝑠 1 2 𝑡 𝑠 𝑡 , and a basis for the solution space is 1 2 1 2 1 0 , 1 2 1 2 0 1 .
  • E The general solution is 𝑠 𝑡 𝑠 + 𝑡 𝑠 𝑡 , and a basis for the solution space is 1 1 1 0 , 1 1 0 1 .

Q8:

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

  • AThe general solution is 𝑠 𝑡 𝑠 + 𝑡 𝑠 𝑡 , and a basis for the solution space is 1 1 1 0 , 1 1 0 1 .
  • B The general solution is 𝑡 𝑠 𝑠 𝑠 𝑡 , and a basis for the solution space is 0 1 1 1 , 1 0 0 1 .
  • CThe general solution is 𝑡 𝑠 + 𝑡 𝑠 𝑠 𝑡 , and a basis for the solution space is 0 1 1 1 , 1 1 0 1 .
  • DThe general solution is 𝑠 𝑡 𝑠 𝑠 𝑡 , and a basis for the solution space is 1 1 1 0 , 1 0 0 1 .
  • EThe general solution is 𝑡 𝑠 𝑠 𝑡 , and a basis for the solution space is 0 1 1 0 , 1 0 0 1 .

Q9:

Let 𝑉 be the space of polynomials in the variable 𝑥 that have degree less than 4. Is 𝑥 + 1 , 𝑥 + 𝑥 + 2 𝑥 , 𝑥 + 𝑥 , 𝑥 + 𝑥 + 𝑥 3 2 2 3 2 a basis for this space?

  • Ayes
  • Bno

Q10:

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

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

Q11:

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

  • A 𝐴 has 𝑚 linearly independent eigenvectors.
  • BThe determinant of 𝐴 is 1.
  • C 𝐴 is idempotent.
  • DThe kernal of 𝐴 is { } 0 .
  • E 𝐴 is a projection matrix.

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