Let and be norms on and respectively. Let be the space of linear transformations from to . Then:
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0 marksMathematicsVector spaces, rank and eigenvaluesNorms, inner products and orthogonalityHardabout 3 min
DSE 2019 Question 30: Vector spaces, rank and eigenvalues
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Let ‖·‖ₙ and ‖·‖ₘ be norms on ℜⁿ and ℜ^m respectively. Let L be the space of linear transformations from ℜⁿ to ℜ^m. Then: (A) Neither ‖·‖* nor ‖·‖** is a norm on L (B) ‖·‖* and ‖·‖** are norms on L (C) ‖L‖*=sup {‖L(x)‖ₘ: x∈ℜⁿ} defines a norm on L (D) ‖L‖**=sup {‖L(x)‖ₘ: x∈ℜⁿ, ‖x‖ₙ≤ 1} defines a norm on L
Also written as: Let | * | _n and | * | _m be norms on Re^n and Re^m respectively. Let L be the space of linear transformations from Re^n to Re^m. Then:
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This question is on Norms, inner products and orthogonality, part of Vector spaces, rank and eigenvalues in Mathematics.
Length and angle in vector spaces and the matrices that preserve them.
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