23.10. The Rank of a Matrix and Linear Independence

Try these first

1. 1.

   Determine the rank of the following matrix: [1 2; 2 4].

   Hint

   Look for redundancy between the rows.

2. 2.

   List conditions when a set of vectors is considered linearly independent.

   Hint

   Think about linear combinations and the zero vector.

3. 3.

   If a matrix has a rank equal to its number of columns, what can we say about its column vectors?

   • They are linearly dependent
   • They are linearly independent
   • They are not related

   Hint

   Recall the definition of rank.

4. 4.

   True or False: A set containing the zero vector can still be linearly independent.

   • True
   • False

   Hint

   Consider the implications of the zero vector in linear combinations.

5. 5.

   Consider a matrix A with vectors [(1, 2), (2, 4), (3, 6), (4, 8)]. What is its rank and what does this imply about the vectors?

   Hint

   Check if any vector can be produced as a linear combination of the others.

6. 6.

   If a system of equations is determined by vectors in R^6, can any matrix formed from them have a rank greater than 6? Justify your answer.

   Hint

   Review the relationship between the rank of a matrix and its dimensional properties.