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Consider the subspace W = {(x1, x2, ..., x10) ∈ R10 : x n = xn–1 + xn–2 for 3 ≤ n ≤ 10} of the vector space R10. The dimension of W is
- a)2
- b)3
- c)9
- d)10
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
Consider the subspace W = {(x1, x2, ..., x10) ∈R10 : x n = xn...
Number of variable is 10 and number of restrictions is 8 so dimension of w is 10-8=2
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Community Answer
Consider the subspace W = {(x1, x2, ..., x10) ∈R10 : x n = xn...
To find the dimension of the subspace W, we need to determine the maximum number of linearly independent vectors that can span W.
Let's analyze the conditions given for the vectors in W:
- The vectors in W are 10-dimensional vectors: (x1, x2, ..., x10).
- The last 8 components of the vector must satisfy the equation xn = xn-1 + xn-2 for n = 3 to 10.
To simplify the analysis, let's consider the last 8 components of the vectors in W as a new vector Z = (z1, z2, ..., z8), where zi = xi+2 for i = 1 to 8. Now, Z is an 8-dimensional vector.
We can rewrite the equation xn = xn-1 + xn-2 as z1 = x3 = x2 + x1. This means that z1 can be expressed as a linear combination of the first two components of the vector (x1, x2), which implies that z1 is dependent on (x1, x2).
Similarly, we can rewrite the equation xn = xn-1 + xn-2 for n = 4 to 10 as zi = xi + xi-1 for i = 2 to 8. This shows that the remaining 7 components of Z are also dependent on (x1, x2).
Therefore, the vector Z (and consequently, the vectors in W) can be expressed as a linear combination of (x1, x2). This implies that the maximum number of linearly independent vectors in W is 2.
Hence, the dimension of W is 2.
Therefore, the correct answer is option 'A' (2).
Let's analyze the conditions given for the vectors in W:
- The vectors in W are 10-dimensional vectors: (x1, x2, ..., x10).
- The last 8 components of the vector must satisfy the equation xn = xn-1 + xn-2 for n = 3 to 10.
To simplify the analysis, let's consider the last 8 components of the vectors in W as a new vector Z = (z1, z2, ..., z8), where zi = xi+2 for i = 1 to 8. Now, Z is an 8-dimensional vector.
We can rewrite the equation xn = xn-1 + xn-2 as z1 = x3 = x2 + x1. This means that z1 can be expressed as a linear combination of the first two components of the vector (x1, x2), which implies that z1 is dependent on (x1, x2).
Similarly, we can rewrite the equation xn = xn-1 + xn-2 for n = 4 to 10 as zi = xi + xi-1 for i = 2 to 8. This shows that the remaining 7 components of Z are also dependent on (x1, x2).
Therefore, the vector Z (and consequently, the vectors in W) can be expressed as a linear combination of (x1, x2). This implies that the maximum number of linearly independent vectors in W is 2.
Hence, the dimension of W is 2.
Therefore, the correct answer is option 'A' (2).
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