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Worksheet: Basis for the Kernel of a Matrix

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 𝑉 be the space of polynomials in the variable π‘₯ that have degree less than 4. Is  π‘₯ + 1 , π‘₯ + π‘₯ + 2 π‘₯ , π‘₯ + π‘₯ , π‘₯ + π‘₯ + π‘₯  3 2 2 3 2 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 .