Math Problem Statement
4minus 3
Solution
To determine the value of for which the given vectors are not linearly independent, we must identify the condition under which the vectors become linearly dependent.
The vectors given in the image are:
Two vectors and are linearly dependent if there exists a scalar such that:
This means:
Expanding this equation, we get two separate equations:
From the second equation:
Substitute into the first equation:
Thus, the vectors are not linearly independent (i.e., they are linearly dependent) when .
Answer: .
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Related Questions:
- How can we determine if three vectors are linearly independent or dependent?
- What is the geometric interpretation of linearly dependent vectors?
- How do we find the determinant of a matrix formed by vectors to check independence?
- What does it mean for vectors to span a vector space?
- How does the concept of linear independence extend to higher dimensions?
Tip: Remember, two vectors are linearly dependent if they lie on the same line through the origin.
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Vectors
Linear Dependence
Formulas
Vector equality in terms of scalar multiplication
Theorems
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Suitable Grade Level
Advanced High School