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The vector [1, 2, 3], [1, 0, 0], [0, 1, 0], [0, 0, 1] are
  • a)
    linearly dependent   
  • b)
    linearly independent
  • c)
    orthogonal
  • d)
    none of these
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
The vector [1, 2, 3], [1, 0, 0], [0, 1, 0], [0, 0, 1] area)linearly de...
Let a = [1, 2, 3], b = [1, 0, 0], c = [0, 1, 0]
 d = [0, 0, 1]
a = b + 2c + 3d
Therefore, vector a, b, c, d are linearly dependent.
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The vector [1, 2, 3], [1, 0, 0], [0, 1, 0], [0, 0, 1] area)linearly de...
**Linear Dependence and Independence**

In linear algebra, vectors are said to be linearly dependent if one or more of the vectors can be expressed as a linear combination of the others. On the other hand, vectors are said to be linearly independent if none of the vectors can be expressed as a linear combination of the others.

**Explanation**

To determine whether the vectors [1, 2, 3], [1, 0, 0], [0, 1, 0], [0, 0, 1] are linearly dependent or independent, we need to check if any vector can be expressed as a linear combination of the others.

If we take the first vector [1, 2, 3] and multiply it by a scalar, for example, 2, we get [2, 4, 6]. However, [2, 4, 6] cannot be expressed as a linear combination of the other vectors.

Similarly, if we take the second vector [1, 0, 0] and multiply it by a scalar, for example, 3, we get [3, 0, 0]. [3, 0, 0] also cannot be expressed as a linear combination of the other vectors.

The same applies to the third vector [0, 1, 0] and the fourth vector [0, 0, 1]. None of these vectors can be expressed as a linear combination of the others.

Since none of the vectors can be expressed as a linear combination of the others, the vectors [1, 2, 3], [1, 0, 0], [0, 1, 0], [0, 0, 1] are linearly independent.

Therefore, the correct answer is option 'b' linearly independent.
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The vector [1, 2, 3], [1, 0, 0], [0, 1, 0], [0, 0, 1] area)linearly dependentb)linearly independentc)orthogonald)none of theseCorrect answer is option 'A'. Can you explain this answer?
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