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If X₁, X2, X3 are linearly independent and K, X, K2 X2 K3 X3 = 0, then a) All K₁, K2, Ka are zero c) K₁, K2, K3 are all equal to 1 b) K₁=0 but K2, K30 d) K₁ K₂=0 but K30?
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If X₁, X2, X3 are linearly independent and K, X, K2 X2 K3 X3 = 0, t...
To understand the given equation and determine the values of K₁, K₂, and K₃, let's break down the problem step by step.
Given:
X₁, X₂, X₃ are linearly independent vectors.
K, K₂, K₃ are constants.
K₁X₁ + K₂X₂ + K₃X₃ = 0
To find the values of K₁, K₂, and K₃, we can consider the following cases:
Case 1: K₁ = 0
If K₁ = 0, then the equation becomes:
0 + K₂X₂ + K₃X₃ = 0
This equation can only hold true if K₂ = 0 and K₃ = 0. Therefore, in this case, K₁ = K₂ = K₃ = 0.
Case 2: K₁ ≠ 0
If K₁ ≠ 0, we can divide the equation by K₁ to isolate X₁:
X₁ + (K₂/K₁)X₂ + (K₃/K₁)X₃ = 0
Case 2.1: K₁ = K₂ = K₃ = 0
If K₁ = K₂ = K₃ = 0, then the equation becomes:
X₁ + 0 + 0 = 0
This implies that X₁ must be equal to zero, which contradicts the given information that X₁ is linearly independent. Therefore, K₁, K₂, and K₃ cannot all be zero.
Case 2.2: K₂ ≠ 0 or K₃ ≠ 0
If K₂ ≠ 0 or K₃ ≠ 0, then for the equation to hold true, X₁ must be a linear combination of X₂ and X₃. However, this contradicts the given information that X₁, X₂, and X₃ are linearly independent. Hence, K₁, K₂, and K₃ cannot be simultaneously non-zero.
Based on the above analysis, we can conclude that the correct answer is:
a) All K₁, K₂, K₃ are zero.
Explanation:
- We have considered two cases: K₁ = 0 and K₁ ≠ 0.
- In Case 1, we found that K₂ and K₃ must also be zero for the equation to hold true.
- In Case 2, we showed that if K₁ ≠ 0, then the equation cannot hold true.
- Therefore, the only possibility is that K₁, K₂, and K₃ are all zero.
Given:
X₁, X₂, X₃ are linearly independent vectors.
K, K₂, K₃ are constants.
K₁X₁ + K₂X₂ + K₃X₃ = 0
To find the values of K₁, K₂, and K₃, we can consider the following cases:
Case 1: K₁ = 0
If K₁ = 0, then the equation becomes:
0 + K₂X₂ + K₃X₃ = 0
This equation can only hold true if K₂ = 0 and K₃ = 0. Therefore, in this case, K₁ = K₂ = K₃ = 0.
Case 2: K₁ ≠ 0
If K₁ ≠ 0, we can divide the equation by K₁ to isolate X₁:
X₁ + (K₂/K₁)X₂ + (K₃/K₁)X₃ = 0
Case 2.1: K₁ = K₂ = K₃ = 0
If K₁ = K₂ = K₃ = 0, then the equation becomes:
X₁ + 0 + 0 = 0
This implies that X₁ must be equal to zero, which contradicts the given information that X₁ is linearly independent. Therefore, K₁, K₂, and K₃ cannot all be zero.
Case 2.2: K₂ ≠ 0 or K₃ ≠ 0
If K₂ ≠ 0 or K₃ ≠ 0, then for the equation to hold true, X₁ must be a linear combination of X₂ and X₃. However, this contradicts the given information that X₁, X₂, and X₃ are linearly independent. Hence, K₁, K₂, and K₃ cannot be simultaneously non-zero.
Based on the above analysis, we can conclude that the correct answer is:
a) All K₁, K₂, K₃ are zero.
Explanation:
- We have considered two cases: K₁ = 0 and K₁ ≠ 0.
- In Case 1, we found that K₂ and K₃ must also be zero for the equation to hold true.
- In Case 2, we showed that if K₁ ≠ 0, then the equation cannot hold true.
- Therefore, the only possibility is that K₁, K₂, and K₃ are all zero.
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If X₁, X2, X3 are linearly independent and K, X, K2 X2 K3 X3 = 0, then a) All K₁, K2, Ka are zero c) K₁, K2, K3 are all equal to 1 b) K₁=0 but K2, K30 d) K₁ K₂=0 but K30?
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If X₁, X2, X3 are linearly independent and K, X, K2 X2 K3 X3 = 0, then a) All K₁, K2, Ka are zero c) K₁, K2, K3 are all equal to 1 b) K₁=0 but K2, K30 d) K₁ K₂=0 but K30? for Electronics and Communication Engineering (ECE) 2024 is part of Electronics and Communication Engineering (ECE) preparation. The Question and answers have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about If X₁, X2, X3 are linearly independent and K, X, K2 X2 K3 X3 = 0, then a) All K₁, K2, Ka are zero c) K₁, K2, K3 are all equal to 1 b) K₁=0 but K2, K30 d) K₁ K₂=0 but K30? covers all topics & solutions for Electronics and Communication Engineering (ECE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If X₁, X2, X3 are linearly independent and K, X, K2 X2 K3 X3 = 0, then a) All K₁, K2, Ka are zero c) K₁, K2, K3 are all equal to 1 b) K₁=0 but K2, K30 d) K₁ K₂=0 but K30?.
If X₁, X2, X3 are linearly independent and K, X, K2 X2 K3 X3 = 0, then a) All K₁, K2, Ka are zero c) K₁, K2, K3 are all equal to 1 b) K₁=0 but K2, K30 d) K₁ K₂=0 but K30? for Electronics and Communication Engineering (ECE) 2024 is part of Electronics and Communication Engineering (ECE) preparation. The Question and answers have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about If X₁, X2, X3 are linearly independent and K, X, K2 X2 K3 X3 = 0, then a) All K₁, K2, Ka are zero c) K₁, K2, K3 are all equal to 1 b) K₁=0 but K2, K30 d) K₁ K₂=0 but K30? covers all topics & solutions for Electronics and Communication Engineering (ECE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If X₁, X2, X3 are linearly independent and K, X, K2 X2 K3 X3 = 0, then a) All K₁, K2, Ka are zero c) K₁, K2, K3 are all equal to 1 b) K₁=0 but K2, K30 d) K₁ K₂=0 but K30?.
Solutions for If X₁, X2, X3 are linearly independent and K, X, K2 X2 K3 X3 = 0, then a) All K₁, K2, Ka are zero c) K₁, K2, K3 are all equal to 1 b) K₁=0 but K2, K30 d) K₁ K₂=0 but K30? in English & in Hindi are available as part of our courses for Electronics and Communication Engineering (ECE).
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