23. Linear Independence

Try these first

1. 1.

   Determine if the following vectors are linearly independent: [1, 2], [2, 3].

   Hint

   Try forming a linear equation with both vectors.

2. 2.

   Are the vectors [1, 0] and [0, 1] linearly independent?

   Hint

   Think about spanning a plane.

3. 3.

   A set of vectors is linearly independent if:

   • At least one vector can be expressed as a linear combination of others
   • The only solution to their linear combination equaling zero is non-trivial
   • The only solution to their linear combination equaling zero is trivial

   Hint

   Recall what a trivial solution is.

4. 4.

   If a set of vectors are collinear in R^2, are they linearly independent?

   • True
   • False

   Hint

   Think about spanning a plane.

5. 5.

   Show that the vectors [1, 0, 0], [0, 1, 0], [0, 0, 1] form a basis for R^3.

   Hint

   Check for linear combinations.

6. 6.

   Given vectors in R^3, [x, 2x, 3x], under what conditions for 'x' are they dependent?

   Hint

   Consider the implications of scalar multiples.