a) Explain if the vectors are linearly dependent. b) Express the vector as a linear combination.
Understand the Problem
The question is asking about the concepts of linear dependence and basis in the context of vectors, specifically how to express a vector as a linear combination of others. This requires understanding of linear algebra principles.
Answer
Determine dependence by solving c1*v1 + c2*v2 + ... = 0. Express via coefficients.
To determine linear dependence, solve for non-zero scalars such that the combination equals zero. If found, the vectors are dependent. For expressing as a linear combination, determine coefficients that express one vector in terms of others.
Answer for screen readers
To determine linear dependence, solve for non-zero scalars such that the combination equals zero. If found, the vectors are dependent. For expressing as a linear combination, determine coefficients that express one vector in terms of others.
More Information
Linear dependence implies at least one vector is a combination of others. The span of vectors might form the entire space or part of it.
Tips
A common mistake is assuming zero vector implies dependence, but this is valid only when not all coefficients are zero.
Sources
- Linear Dependence and Span - math.ryerson.ca
- 2.4: Linear independence - Mathematics LibreTexts - math.libretexts.org
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