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Let P ≠ 0 be a 3 × 3 real matrix. There exist linearly independent vectors x and y such that Px = 0 and Py = 0. The dimension of the range space of P is
- a)0
- b)1
- c)2
- d)3
Correct answer is option 'B'. Can you explain this answer?
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Let P ≠ 0 be a 3 × 3 real matrix. There exist linearly indepe...
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Let P ≠ 0 be a 3 × 3 real matrix. There exist linearly indepe...
Explanation:
Let's assume that the rank of the matrix P is r.
Since we know that x and y are linearly independent vectors and Px=0 and Py=0, this implies that x and y both are in the null space of P.
Hence, the nullity of P is at least 2 (since there are at least two linearly independent vectors in the null space of P).
Since the rank of P is r and the nullity of P is at least 2, we have
r + nullity of P = 3
=> r + 2 >= 3
=> r >= 1
This means that the rank of P is at least 1.
However, we know that Px=0 and hence, x is in the null space of P. Since x is non-zero, this implies that the nullity of P is at least 1.
So, we have r >= 1 and nullity of P >= 1. This implies that the range space of P has dimension at most 2 (since the sum of rank and nullity is at most the dimension of the matrix).
However, since we know that x and y are linearly independent vectors, this implies that the nullity of P is exactly 1 (since the null space of P cannot contain more than one linearly independent vector). Hence, the rank of P is exactly 2 and the range space of P has dimension 1.
Hence, the correct option is (B).
Let's assume that the rank of the matrix P is r.
Since we know that x and y are linearly independent vectors and Px=0 and Py=0, this implies that x and y both are in the null space of P.
Hence, the nullity of P is at least 2 (since there are at least two linearly independent vectors in the null space of P).
Since the rank of P is r and the nullity of P is at least 2, we have
r + nullity of P = 3
=> r + 2 >= 3
=> r >= 1
This means that the rank of P is at least 1.
However, we know that Px=0 and hence, x is in the null space of P. Since x is non-zero, this implies that the nullity of P is at least 1.
So, we have r >= 1 and nullity of P >= 1. This implies that the range space of P has dimension at most 2 (since the sum of rank and nullity is at most the dimension of the matrix).
However, since we know that x and y are linearly independent vectors, this implies that the nullity of P is exactly 1 (since the null space of P cannot contain more than one linearly independent vector). Hence, the rank of P is exactly 2 and the range space of P has dimension 1.
Hence, the correct option is (B).
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Let P ≠ 0 be a 3 × 3 real matrix. There exist linearly independent vectors x and y such that Px = 0 and Py = 0. The dimension of the range space of P isa)0b)1c)2d)3Correct answer is option 'B'. Can you explain this answer?
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