Illustrative Examples: When Matrix Rank Confirms Linear Dependence

📚 Quick Study Guide

• 🔢 The rank of a matrix is the number of linearly independent rows or columns.
• 🧑‍🏫 If the rank of a matrix is less than the number of its columns, the columns are linearly dependent.
• 📝 For a matrix $A$ with $n$ columns, if $rank(A) < n$, the columns of $A$ are linearly dependent.
• 💡 Linear dependence means that at least one vector in the set can be written as a linear combination of the others.
• 🔍 If a matrix has a row or column of zeros, or if one row/column is a multiple of another, the columns are linearly dependent, and the rank is less than the number of columns.

🧪 Practice Quiz

1. Which of the following statements is true regarding matrix rank and linear dependence?
   1. A) A matrix with full rank always has linearly dependent columns.
   2. B) If the rank of a matrix equals the number of columns, the columns are linearly dependent.
   3. C) If the rank of a matrix is less than the number of columns, the columns are linearly dependent.
   4. D) Matrix rank has no relation to linear dependence.
2. Given a 3x3 matrix A, if rank(A) = 2, what can you conclude about its columns?
   1. A) The columns are linearly independent.
   2. B) The columns are linearly dependent.
   3. C) The columns are orthonormal.
   4. D) The columns form a basis for $R^3$.
3. Consider a 4x2 matrix B. What is the maximum possible rank of B, and what does it imply about the linear dependence of its columns when the rank is at its maximum?
   1. A) Max rank = 4; columns are linearly independent.
   2. B) Max rank = 2; columns are linearly independent.
   3. C) Max rank = 4; columns are linearly dependent.
   4. D) Max rank = 2; columns are linearly dependent.
4. If a 5x5 matrix C has a row of all zeros, what can be said about its rank and the linear dependence of its columns?
   1. A) rank(C) = 5; columns are linearly independent.
   2. B) rank(C) = 5; columns are linearly dependent.
   3. C) rank(C) < 5; columns are linearly independent.
   4. D) rank(C) < 5; columns are linearly dependent.
5. Suppose you have a 3x3 matrix where one column is a scalar multiple of another. What does this indicate about the matrix's rank?
   1. A) The rank is 3.
   2. B) The rank is greater than 3.
   3. C) The rank is less than 3.
   4. D) The rank is equal to the number of rows.
6. A 4x4 matrix D has a rank of 3. How many linearly independent columns does D have?
   1. A) 0
   2. B) 1
   3. C) 3
   4. D) 4
7. Which scenario guarantees that the columns of a matrix are linearly dependent?
   1. A) The matrix is square.
   2. B) The determinant of the matrix is non-zero.
   3. C) The matrix has more columns than rows.
   4. D) All entries in the matrix are positive.