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Linear Algebra: Why Some Arrows Never Turn

Systems of equations, matrices, determinants, vector spaces, transformations, eigenvectors and least squares: the geometry behind graphics, search and data science.

Linear Algebra · 50 questions

Try 3 questions

1. Two linear equations in two unknowns describe two lines. What happens if the lines are parallel and distinct?

  1. There is no solution
  2. There are infinitely many
  3. There is exactly one solution
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Answer: There is no solution

A solution is a point on both lines; distinct parallel lines never meet, so the system is inconsistent.

2. How many solutions can a system of linear equations have?

  1. Always exactly one solution
  2. None, one, or infinitely many
  3. Any whole number, such as two
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Answer: None, one, or infinitely many

If two different solutions exist, every point on the line through them is also a solution, so 'exactly two' is impossible.

3. What does Gaussian elimination do to a system of equations?

  1. Reduces it to triangular form
  2. Squares each equation first
  3. Guesses values and checks them
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Answer: Reduces it to triangular form

Row operations eliminate variables step by step until back-substitution can read off the solution.

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