23.7. Examples

Try these first

1. 1.

   Are the vectors [2, 1], [4, 2] linearly independent?

   Hint

   Check if one vector can be expressed as a scalar multiple of another.

2. 2.

   Is the set {(1, 0), (0, 1)} linearly independent?

   Hint

   Consider the geometric representation of these vectors.

3. 3.

   If a set of three vectors in R3 is dependent, what can be said about their relationships?

   Hint

   Recall the definitions of linear independence and dependence.

4. 4.

   Is the following statement true or false? 'Three vectors can be linearly independent in a 2-D space.'

   • True
   • False

   Hint

   Consider the dimensionality of the space.

5. 5.

   Given the vectors [2, 3, 1], [4, 6, 2], [5, 7, 4], determine if they are linearly independent or dependent and explain your reasoning.

   Hint

   Form a matrix and use row reduction.

6. 6.

   Using the vectors (1, 2, -1), (2, 4, 2), (3, 6, 3), show they are dependent and find the relationship between them.

   Hint

   Look for a common scalar that relates the vectors.