23.8. Properties of Linearly Independent Sets

Try these first

1. 1.

   Determine if the set { [1, 0], [0, 1], [1, 1] } is linearly independent.

   Hint

   Look for combinations of vectors that produce another vector in the set.

2. 2.

   Is the set { [0, 0], [1, 1] } linearly independent?

   Hint

   Remember the importance of the zero vector.

3. 3.

   A set containing the zero vector is linearly dependent.

   • True
   • False

   Hint

   Consider what the zero vector represents in a linear combination.

4. 4.

   In R^4, a set of five vectors must be linearly independent.

   • True
   • False

   Hint

   Reflect on the dimensionality of R^n.

5. 5.

   Prove that the three vectors v1 = (1,0,0), v2 = (0,1,0), v3 = (0,0,1) are linearly independent.

   Hint

   You can set up a matrix and check its rank or use geometric reasoning.

6. 6.

   Consider the vectors u1 = (1,1,0), u2 = (2,2,0), u3 = (0,3,0). Are these vectors linearly independent? Justify your answer.

   Hint

   Check if any vector is a linear combination of the others.