Linearly independent refers to a set of vectors in a vector space where no vector can be expressed as a linear combination of the others. This means that each vector adds a unique dimension to the space, and the only way to create the zero vector from this set is by scaling all vectors by zero. The concept of linear independence is crucial because it helps to determine the basis of a vector space, which is essential for understanding its structure and dimensionality.
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If a set of vectors is linearly independent, then the number of vectors in that set cannot exceed the dimension of the vector space.
To test for linear independence, you can form a matrix with the vectors as columns and determine if its determinant is non-zero.
If at least one vector in a set can be expressed as a linear combination of others, then the set is linearly dependent.
The zero vector is always linearly dependent with any set of vectors, as it can be represented by multiplying all vectors by zero.
In $ ext{R}^n$, any set of more than n vectors is guaranteed to be linearly dependent.
Review Questions
How can you determine if a set of vectors is linearly independent or dependent?
To determine if a set of vectors is linearly independent, you can arrange the vectors as columns in a matrix and calculate its determinant. If the determinant is non-zero, this indicates that the vectors are linearly independent. Alternatively, you can set up an equation where a linear combination of these vectors equals the zero vector and check if the only solution is when all coefficients are zero. If there are other solutions, then the vectors are dependent.
Discuss the importance of linear independence in relation to the basis of a vector space.
Linear independence is vital for establishing a basis in a vector space because a basis consists of linearly independent vectors that span the entire space. This means each vector contributes uniquely to representing any point in that space. Without linear independence, some vectors would not add new information and could be removed from the basis, resulting in redundancy. Therefore, identifying linearly independent sets helps in forming an efficient and minimal representation of vector spaces.
Evaluate how linear independence affects the solutions to systems of linear equations.
The concept of linear independence directly influences the nature of solutions in systems of linear equations. If the columns of the coefficient matrix are linearly independent, it implies that there will be a unique solution to the system. Conversely, if they are dependent, it indicates either no solutions or infinitely many solutions exist. Understanding this relationship between linear independence and solutions helps in analyzing systems effectively, allowing for predictions about their behavior based on their vector representations.
A basis of a vector space is a set of vectors that are linearly independent and span the entire space, meaning any vector in the space can be expressed as a linear combination of the basis vectors.
Span: The span of a set of vectors is the collection of all possible linear combinations that can be formed from those vectors, essentially covering all points that can be reached in the vector space.