Algorithm for testing linear independence of vectors

📚 Quick Study Guide

🔍 Vectors are linearly independent if no non-trivial linear combination of them equals the zero vector. 💡 To test for linear independence, set up a homogeneous system of equations using the vectors as columns in a matrix. 📝 Solve the system. If the only solution is the trivial solution (all variables are zero), the vectors are linearly independent. ➗ If there are non-trivial solutions (at least one variable can be non-zero), the vectors are linearly dependent. 📌 The determinant of a square matrix formed by the vectors can be used: non-zero determinant implies linear independence. 📐 For two vectors, linear independence simply means they are not scalar multiples of each other. 🧭 A set of vectors containing the zero vector is always linearly dependent.

Practice Quiz

1. Which of the following statements is true regarding linearly independent vectors?
   1. Vectors are linearly independent if one vector can be written as a scalar multiple of another.
   2. Vectors are linearly independent if their linear combination always results in the zero vector.
   3. Vectors are linearly independent if the only solution to their homogeneous equation is the trivial solution.
   4. Vectors are linearly independent if they span a subspace of dimension less than the number of vectors.
2. Consider the vectors $v_1 = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$ and $v_2 = \begin{bmatrix} 2 \\ 4 \end{bmatrix}$. Are they linearly independent?
   1. Yes, they are linearly independent.
   2. No, they are linearly dependent.
   3. It cannot be determined.
   4. Only if the determinant is non-zero.
3. Given the matrix $A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, what can be said about its column vectors?
   1. They are linearly dependent.
   2. They are linearly independent.
   3. They are orthogonal.
   4. They span $\mathbb{R}^1$.
4. If a set of vectors contains the zero vector, what can you conclude?
   1. The set is linearly independent.
   2. The set is linearly dependent.
   3. The set could be either linearly independent or dependent, depending on the other vectors.
   4. The set spans the entire vector space.
5. What is the determinant of a matrix whose column vectors are linearly dependent?
   1. 1
   2. -1
   3. 0
   4. Undefined
6. For what value of $k$ are the vectors $\begin{bmatrix} 1 \\ 2 \end{bmatrix}$ and $\begin{bmatrix} 2 \\ k \end{bmatrix}$ linearly dependent?
   1. $k = 1$
   2. $k = 2$
   3. $k = 3$
   4. $k = 4$
7. Which of the following is a correct method to check for linear independence of a set of vectors?
   1. Check if all vectors are orthogonal to each other.
   2. Calculate the dot product of all pairs of vectors.
   3. Form a matrix with the vectors as columns and check if the determinant is zero.
   4. Verify that each vector has a magnitude of 1.