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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?
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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.
d = [0, 0, 1]
a = b + 2c + 3d
Therefore, vector a, b, c, d are linearly dependent.
Most Upvoted Answer
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.
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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