91Ó°ÊÓ

A linear combination is an expression that multiplies scalars by vectors before adding them together. This is central in understanding vector space theory. For our problem, vectors \(v_1\), \(v_2\), and \(v_3\) can be expressed as linear combinations of each other.
Consider a scenario where you have three distinct vectors, \(v_1 = (0, 3, 1, -1)\), \(v_2 = (6, 0, 5, 1)\), and \(v_3 = (4, -7, 1, 3)\).
To find a linear combination of these vectors, we want to express one vector as a sum of the other two. For example, expressing \(v_1\) as a linear combination of \(v_2\) and \(v_3\) involves finding scalars \(a\) and \(b\) such that:

• \(v_1 = a \cdot v_2 + b \cdot v_3\)

This approach is useful because it demonstrates how vectors relate to each other within a vector space.
In technical terms, finding a valid linear combination where not all scalars are zero indicates the vectors are linearly dependent. In our exercise, \(v_1\) can be expressed as \(-\frac{2}{3}v_2 + \frac{7}{3}v_3\), confirming linear dependence.

A vector space is a fundamental concept in linear algebra. It refers to a collection of vectors that can be added together and multiplied by scalars to produce another vector within the same space.
Essentially, a vector space encompasses the set of operations (addition, scalar multiplication) that work together in coherent ways under certain rules, like the commutative and associative laws.
The vectors \(v_1, v_2,\) and \(v_3\) from the exercise are elements within \(\mathbb{R}^4\), a four-dimensional real vector space, denoted \(\mathbb{R}^n\) where \(n\) indicates the number of dimensions.
These vectors demonstrate the capability to be combined into a new vector within this space. When these vectors are linearly dependent, it shows that one vector can be formed by a combination of others, highlighting the interconnectedness typically observed within vector spaces.

• If a vector space only contains the zero vector or no vector can be written as a combination of others, then its basis vectors are independent.
• It's crucial to remember that vector spaces cover infinite possibilities for combinations but are constrained by their governing laws.

Solving linear dependence requires setting up a system of equations. A system of equations involves finding values of variables that satisfy multiple conditions represented by equalities.
In our exercise, to determine the linear dependence of \(v_1, v_2,\) and \(v_3\), we transformed the dependency equation:

• \(c_1v_1 + c_2v_2 + c_3v_3 = 0\)

into a system of equations.
This means solving for scalars \(c_1, c_2,\) and \(c_3\) using the entries from each vector:

• \(0c_1 + 6c_2 + 4c_3 = 0\)
• \(3c_1 + 0c_2 - 7c_3 = 0\)
• \(c_1 + 5c_2 + c_3 = 0\)
• \(-c_1 + c_2 + 3c_3 = 0\)

This set of linear equations is then solved using methods like substitution or elimination. In our scenario, the solution \(c_1 = \frac{7}{3}c_3\), \(c_2 = -\frac{2}{3}c_3\), with \(c_3\) being a free parameter confirms that these vectors are linearly dependent because their solution allows non-trivial values. The ability to solve such systems is vital for analyzing linear relationships between vectors.