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 012101125𝑥𝑦𝑧=000.

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

Q2:

Find the general solution of the following system of linear equations: 012121145𝑥𝑦𝑧=000, and then find a basis for its solution space.

  • AThe general solution is 4𝑡2𝑡𝑡. A basis for the solution space is 421.
  • BThe general solution is 3𝑡2𝑡𝑡. A basis for the solution space is 321.
  • CThe general solution is 𝑡2𝑡3𝑡. A basis for the solution space is 123.
  • DThe general solution is 3𝑡2𝑡𝑡. A basis for the solution space is 123.
  • EThe general solution is 𝑡2𝑡4𝑡. A basis for the solution space is 124.

Q3:

Suppose that 𝐴 and 𝐵 are 𝑚×𝑚 matrices and v is a vector in the nullspace of 𝐴. Consider this statement: 𝐴𝐵=.v0 Which of the following is true?

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

Q4:

Find the general solution of the system of linear equations 1101111031123303𝑥𝑦𝑧𝑤=0000, and hence find a basis for its solution space.

  • AThe general solution is 12𝑠12𝑡12𝑠12𝑡𝑠𝑡, and a basis for the solution space is 121210,121201.
  • BThe general solution is 𝑡12𝑠12𝑡𝑠12𝑠12𝑡, and a basis for the solution space is 012112,112012.
  • CThe general solution is 𝑠𝑡𝑠𝑡𝑠𝑡, and a basis for the solution space is 1110,1101.
  • DThe general solution is 𝑡𝑠𝑡𝑠𝑠𝑡, and a basis for the solution space is 0111,1101.
  • EThe general solution is 𝑠𝑡𝑠+𝑡𝑠𝑡, and a basis for the solution space is 1110,1101.

Q5:

Find the general solution of the system of linear equations 1101211210110000𝑥𝑦𝑧𝑤=0000, and hence find a basis for its solution space.

  • AThe general solution is 𝑠𝑡𝑠+𝑡𝑠𝑡, and a basis for the solution space is 1110,1101.
  • BThe general solution is 𝑠𝑡𝑠𝑠𝑡, and a basis for the solution space is 1110,1001.
  • CThe general solution is 𝑡𝑠+𝑡𝑠𝑠𝑡, and a basis for the solution space is 0111,1101.
  • DThe general solution is 𝑡𝑠𝑠𝑠𝑡, and a basis for the solution space is 0111,1001.
  • EThe general solution is 𝑡𝑠𝑠𝑡, and a basis for the solution space is 0110,1001.

Q6:

Find the general solution to the system of linear equations 1101211210110111𝑥𝑦𝑧𝑤=0000 and hence find a basis for its solution space.

  • AThe general solution is 𝑡𝑡𝑡0. A basis for the solution space is 0111.
  • BThe general solution is 0𝑡𝑡𝑡. A basis for the solution space is 0111.
  • CThe general solution is 𝑡𝑡𝑡0. A basis for the solution space is 1110.
  • DThe general solution is 𝑡2𝑡𝑡0. A basis for the solution space is 1210.
  • EThe general solution is 0𝑡2𝑡𝑡. A basis for the solution space is 0121.

Q7:

Find the general solution of the following system of linear equations: 112121345𝑥𝑦𝑧=000, and then find a basis for its solution space.

  • AThe general solution is 3𝑡2𝑡4𝑡. A basis for the solution space is 324.
  • BThe general solution is 𝑡𝑡3𝑡. A basis for the solution space is 113.
  • CThe general solution is 3𝑡𝑡𝑡. A basis for the solution space is 311.
  • DThe general solution is 4𝑡2𝑡3𝑡. A basis for the solution space is 423.
  • EThe general solution is 3𝑡𝑡𝑡. A basis for the solution space is 113.

Q8:

Find the general solution of the system of linear equations 1011111031323303𝑥𝑦𝑧𝑤=0000, and then find a basis for its solution space.

  • AThe general solution is 0𝑡𝑡𝑡. A basis for the solution space is 0111.
  • BThe general solution is 𝑡𝑡3𝑡3𝑡. A basis for the solution space is 1133.
  • CThe general solution is 0𝑡𝑡𝑡. A basis for the solution space is 1110.
  • DThe general solution is 𝑡𝑡𝑡0. A basis for the solution space is 1110.
  • EThe general solution is 𝑡𝑡3𝑡3𝑡. A basis for the solution space is 1133.

Q9:

Find the general solution of the following system of linear equations: 112120344𝑥𝑦𝑧=000, and then find a basis for its solution space.

  • AThe general solution is 3𝑡𝑡𝑡. A basis for the solution space is 311.
  • BThe general solution is 4𝑡2𝑡𝑡. A basis for the solution space is 421.
  • CThe general solution is 𝑡2𝑡4𝑡. A basis for the solution space is 124.
  • DThe general solution is 4𝑡2𝑡𝑡. A basis for the solution space is 124.
  • EThe general solution is 𝑡𝑡3𝑡. A basis for the solution space is 113.

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