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The vectors (1,0,0) (0,1,0) (0,0,1) (1,1,1) are: Linearly dependent Linearly independent Both None of these Answer is linearly independent. Please explain how?
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The vectors (1,0,0) (0,1,0) (0,0,1) (1,1,1) are: Linearly dependent Li...
Linear Independence of Vectors
Definition: A set of vectors is said to be linearly independent if no vector in the set can be expressed as a linear combination of the other vectors.
Analysis of Given Vectors
Let's examine the given vectors:
v1 = (1,0,0)
v2 = (0,1,0)
v3 = (0,0,1)
v4 = (1,1,1)
Linear Combination of Vectors
We can check whether these vectors are linearly independent by checking if there exists a non-trivial linear combination of these vectors that equals the zero vector.
Let's assume that:
a1v1 + a2v2 + a3v3 + a4v4 = 0
where a1, a2, a3, a4 are not all zero.
This equation can be written as:
a1(1,0,0) + a2(0,1,0) + a3(0,0,1) + a4(1,1,1) = (0,0,0)
which simplifies to:
(a1 + a4, a2 + a4, a3 + a4) = (0,0,0)
This system of linear equations has only one solution, a1 = a2 = a3 = -a4.
Therefore, the only linear combination of these vectors that equals the zero vector is:
-v1 - v2 - v3 + v4 = (0,0,0)
Since this is a trivial linear combination, the given vectors are linearly independent.
Conclusion
The vectors (1,0,0), (0,1,0), (0,0,1), and (1,1,1) are linearly independent.
Community Answer
The vectors (1,0,0) (0,1,0) (0,0,1) (1,1,1) are: Linearly dependent Li...
One important characteristic of linearly dependent vectors is that they are coplanar. Here, as you can see, these 4 points by no means lie on a single plane. Hence, linearly independent.
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The vectors (1,0,0) (0,1,0) (0,0,1) (1,1,1) are: Linearly dependent Linearly independent Both None of these Answer is linearly independent. Please explain how?
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