Which of the following sets of vectors are linearly independent? This is a common question in linear algebra, a branch of mathematics that deals with vector spaces and linear transformations. Linear independence is a crucial concept in understanding the properties of vectors and their relationships. In this article, we will explore the concept of linear independence and analyze different sets of vectors to determine their linear independence.
Linear independence refers to the property of a set of vectors where no vector in the set can be expressed as a linear combination of the other vectors. In other words, if a set of vectors is linearly independent, then each vector in the set must be unique and cannot be derived from the others. This property is essential in various applications, such as solving systems of linear equations, finding the basis of a vector space, and analyzing the dimension of a vector space.
Let’s consider the following sets of vectors and determine their linear independence:
1. Set A: {v1 = (1, 2), v2 = (3, 4), v3 = (5, 6)}
2. Set B: {w1 = (1, 0, 0), w2 = (0, 1, 0), w3 = (0, 0, 1)}
3. Set C: {x1 = (1, 1, 1), x2 = (2, 2, 2), x3 = (3, 3, 3)}
4. Set D: {y1 = (1, 2, 3), y2 = (4, 5, 6), y3 = (7, 8, 9)}
To determine the linear independence of each set, we can use the concept of the determinant. If the determinant of the matrix formed by the vectors in a set is non-zero, then the set is linearly independent. Otherwise, the set is linearly dependent.
For Set A, the matrix formed by the vectors is:
| 1 3 5 |
| 2 4 6 |
The determinant of this matrix is (1 4) – (3 2) = 4 – 6 = -2, which is non-zero. Therefore, Set A is linearly independent.
For Set B, the matrix formed by the vectors is:
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
The determinant of this matrix is 1, which is non-zero. Hence, Set B is linearly independent.
For Set C, the matrix formed by the vectors is:
| 1 2 3 |
| 1 2 3 |
| 1 2 3 |
The determinant of this matrix is 0, which indicates that Set C is linearly dependent. This is because each vector in Set C is a scalar multiple of the other vectors.
For Set D, the matrix formed by the vectors is:
| 1 4 7 |
| 2 5 8 |
| 3 6 9 |
The determinant of this matrix is (1 5 9) – (4 6 3) = 45 – 72 = -27, which is non-zero. Therefore, Set D is linearly independent.
In conclusion, the linear independence of a set of vectors can be determined by analyzing the determinant of the matrix formed by the vectors. Sets A, B, and D are linearly independent, while Set C is linearly dependent. Understanding the concept of linear independence is vital in linear algebra and has numerous applications in various fields of mathematics and engineering.
