23.6. Matrix Approach: Row Reduction

Try these first

1. 1.

   Determine whether the vectors [1, 2] and [2, 4] are linearly independent.

   Hint

   Check if one vector can be expressed as a multiple of the other.

2. 2.

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

   Hint

   Visualize their placement on a graph.

3. 3.

   What is the first step to determine if vectors are linearly independent using a matrix?

   • A. Check their dot product
   • B. Form a matrix with them as columns
   • C. Look for scalar multiples

   Hint

   Think about the initial setup in matrix terms.

4. 4.

   True or False: If the number of pivot columns equals the number of vectors, they are dependent.

   • True
   • False

   Hint

   Recall the defining characteristics of dependence.

5. 5.

   Given the vectors A: [1, 2, 3], B: [4, 5, 6], and C: [7, 8, 9], determine if they are independent using row reduction.

   Hint

   Look for identical rows during reduction.

6. 6.

   In R^3, analyze the vectors [1, 2, 3], [2, 4, 6], [1, 0, 0]. Is it possible for them to be independent?

   Hint

   Check whether any vector can be generated from others.