Self-Assessment: Are Your Vectors Linearly Independent? College Level Quiz.

📚 Quick Study Guide

🔍 Vectors are linearly independent if no vector in the set can be written as a linear combination of the others. ➕ A set of vectors {$v_1, v_2, ..., v_n$} is linearly independent if the equation $c_1v_1 + c_2v_2 + ... + c_nv_n = 0$ has only the trivial solution ($c_1 = c_2 = ... = c_n = 0$). 🧑‍🏫 To check for linear independence, form a matrix with the vectors as columns and row reduce to echelon form. If there is a pivot in every column, the vectors are linearly independent. 📐 If you have more vectors than the dimension of the vector space, the vectors are linearly dependent. 💡 A set containing only one non-zero vector is always linearly independent. ✍️ Two vectors are linearly independent if neither is a scalar multiple of the other.

🧪 Practice Quiz

1. Question 1: Which of the following sets of vectors is linearly independent?
1. {$[1, 2], [2, 4]$}
2. {$[1, 0], [0, 1]$}
3. {$[1, 1], [2, 2]$}
4. {$[1, -1], [-1, 1]$}
2. Question 2: Determine if the following vectors are linearly independent: $[1, 0, 0], [0, 1, 0], [0, 0, 1]$
1. Linearly Dependent
2. Linearly Independent
3. Cannot be determined
4. The question is ambiguous
3. Question 3: For what value(s) of $k$ are the vectors $[1, 2]$ and $[k, 4]$ linearly dependent?
1. $k = 1$
2. $k = 2$
3. $k = 3$
4. $k = 4$
4. Question 4: Which statement is true about a set of vectors that includes the zero vector?
1. It is always linearly independent.
2. It is always linearly dependent.
3. It can be either linearly independent or dependent, depending on the other vectors.
4. It is impossible to determine.
5. Question 5: Are the vectors $[1, 2, 3], [0, 1, 4], [0, 0, 1]$ linearly independent?
1. Yes
2. No
3. Cannot be determined
4. Insufficient information
6. Question 6: If a matrix has a row of zeros, what does this imply about its column vectors?
1. They are linearly independent.
2. They are linearly dependent.
3. They might be linearly independent or dependent.
4. The question is unanswerable.
7. Question 7: Which of the following pairs of vectors are linearly independent?
1. $[2,4]$ and $[3,6]$
2. $[1,0]$ and $[1,1]$
3. $[5,5]$ and $[-1,-1]$
4. $[a,b]$ and $[2a, 2b]$