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Lesson: Basis for the Kernel of a Matrix

In this lesson, we will learn how to find the general solution of a system of linear equations and determine a basis for its solution space.

Worksheet • 11 Questions

Q1:

Find a basis for the solution space of the system

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

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

Q3:

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

  • AThe 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 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ .
  • B 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 ⎞ ⎟ ⎟ ⎠ ⎫ ⎬ ⎭ .
  • C 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 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ .
  • DThe 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 ⎞ ⎟ ⎟ ⎠ ⎫ ⎬ ⎭ .
  • EThe 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 ⎞ ⎟ ⎟ ⎠ ⎫ ⎬ ⎭ .

Q4:

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 𝑠 𝑡 , and a basis for the solution space is ⎧ ⎨ ⎩ ⎛ ⎜ ⎜ ⎝ − 1 1 1 0 ⎞ ⎟ ⎟ ⎠ , ⎛ ⎜ ⎜ ⎝ − 1 0 0 1 ⎞ ⎟ ⎟ ⎠ ⎫ ⎬ ⎭ .
  • BThe general solution is 𝑡 𝑠 𝑠 𝑡 , and a basis for the solution space is ⎧ ⎨ ⎩ ⎛ ⎜ ⎜ ⎝ 0 1 1 0 ⎞ ⎟ ⎟ ⎠ , ⎛ ⎜ ⎜ ⎝ 1 0 0 1 ⎞ ⎟ ⎟ ⎠ ⎫ ⎬ ⎭ .
  • C 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 ⎞ ⎟ ⎟ ⎠ ⎫ ⎬ ⎭ .
  • DThe 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 ⎞ ⎟ ⎟ ⎠ ⎫ ⎬ ⎭ .
  • EThe 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 ⎞ ⎟ ⎟ ⎠ ⎫ ⎬ ⎭ .

Q5:

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?

  • Ano
  • Byes

Q6:

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

  • AThe general solution is ⎛ ⎜ ⎜ ⎝ − 𝑡 𝑡 𝑡 0 ⎞ ⎟ ⎟ ⎠ . A basis for the solution space is ⎧ ⎨ ⎩ ⎛ ⎜ ⎜ ⎝ − 1 1 1 0 ⎞ ⎟ ⎟ ⎠ ⎫ ⎬ ⎭ .
  • BThe general solution is ⎛ ⎜ ⎜ ⎝ − 𝑡 𝑡 𝑡 0 ⎞ ⎟ ⎟ ⎠ . A basis for the solution space is ⎧ ⎨ ⎩ ⎛ ⎜ ⎜ ⎝ 0 1 1 − 1 ⎞ ⎟ ⎟ ⎠ ⎫ ⎬ ⎭ .
  • CThe general solution is ⎛ ⎜ ⎜ ⎝ 0 𝑡 𝑡 − 𝑡 ⎞ ⎟ ⎟ ⎠ . A basis for the solution space is ⎧ ⎨ ⎩ ⎛ ⎜ ⎜ ⎝ 0 1 1 − 1 ⎞ ⎟ ⎟ ⎠ ⎫ ⎬ ⎭ .
  • DThe general solution is ⎛ ⎜ ⎜ ⎝ 0 𝑡 2 𝑡 𝑡 ⎞ ⎟ ⎟ ⎠ . A basis for the solution space is ⎧ ⎨ ⎩ ⎛ ⎜ ⎜ ⎝ 0 1 2 1 ⎞ ⎟ ⎟ ⎠ ⎫ ⎬ ⎭ .
  • EThe general solution is ⎛ ⎜ ⎜ ⎝ 𝑡 2 𝑡 𝑡 0 ⎞ ⎟ ⎟ ⎠ . A basis for the solution space is ⎧ ⎨ ⎩ ⎛ ⎜ ⎜ ⎝ 1 2 1 0 ⎞ ⎟ ⎟ ⎠ ⎫ ⎬ ⎭ .

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

Q8:

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

  • AThe general solution is ⎛ ⎜ ⎜ ⎝ 0 − 𝑡 − 𝑡 𝑡 ⎞ ⎟ ⎟ ⎠ . A basis for the solution space is ⎧ ⎨ ⎩ ⎛ ⎜ ⎜ ⎝ 0 − 1 − 1 1 ⎞ ⎟ ⎟ ⎠ ⎫ ⎬ ⎭ .
  • BThe general solution is ⎛ ⎜ ⎜ ⎝ 0 − 𝑡 − 𝑡 𝑡 ⎞ ⎟ ⎟ ⎠ . A basis for the solution space is ⎧ ⎨ ⎩ ⎛ ⎜ ⎜ ⎝ 1 − 1 − 1 0 ⎞ ⎟ ⎟ ⎠ ⎫ ⎬ ⎭ .
  • CThe general solution is ⎛ ⎜ ⎜ ⎝ 𝑡 − 𝑡 − 𝑡 0 ⎞ ⎟ ⎟ ⎠ . A basis for the solution space is ⎧ ⎨ ⎩ ⎛ ⎜ ⎜ ⎝ 1 − 1 − 1 0 ⎞ ⎟ ⎟ ⎠ ⎫ ⎬ ⎭ .
  • DThe general solution is ⎛ ⎜ ⎜ ⎝ 𝑡 − 𝑡 3 𝑡 − 3 𝑡 ⎞ ⎟ ⎟ ⎠ . A basis for the solution space is ⎧ ⎨ ⎩ ⎛ ⎜ ⎜ ⎝ 1 − 1 3 − 3 ⎞ ⎟ ⎟ ⎠ ⎫ ⎬ ⎭ .
  • EThe general solution is ⎛ ⎜ ⎜ ⎝ 𝑡 − 𝑡 3 𝑡 3 𝑡 ⎞ ⎟ ⎟ ⎠ . A basis for the solution space is ⎧ ⎨ ⎩ ⎛ ⎜ ⎜ ⎝ 1 − 1 3 3 ⎞ ⎟ ⎟ ⎠ ⎫ ⎬ ⎭ .

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

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?

  • AIf 𝐵 is a multiple of the identity matrix, then the statement is true.
  • BIf 𝐵 is a multiple of the identity matrix, then the statement is false.
  • CIf the statement is true, then 𝐵 is a multiple of the identity matrix.
  • DThe statement is always true.
  • EThe statement is true if and only if 𝐵 is a multiple of the identity matrix.

Q11:

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

  • AThe kernal of 𝐴 is { } 0 .
  • B 𝐴 is a projection matrix.
  • CThe determinant of 𝐴 is 1.
  • D 𝐴 is idempotent.
  • E 𝐴 has 𝑚 linearly independent eigenvectors.
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