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terla in this subsection allows us to express a geometric relationship that we have not yet seen between the range space and the null space of a linear map. (a) Represent f: R³ → R given by (3) V3 with respect to the standard bases and show that → 1v₁ + 2v₂ + 3v3

terla in this subsection allows us to express a geometric relationship that we have not yet seen between the range space and the null space of a linear map. (a) Represent f: R³ → R given by (3) V3 with respect to the standard bases and show that → 1v₁ + 2v₂ + 3v3

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.4: Linear Transformations
Problem 24EQ
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Please do Exercise 3.27 part A,B,C,D and please show step by step and explain

✓ 3.27 The material in this subsection allows us to express a geometric relationship that we have not yet seen between the range space and the null space of a linear map. (a) Represent f: R³ → R given by VI V2 V3. with respect to the standard bases and show that (-) 2 is a member of the perp of the null space. Prove that (f) is equal to the span of this vector. → 1v₁ + 2v₂ + 3v3 (b) Generalize that to apply to any f: R" → R. (c) Represent f: R³ → R² V1 V2 V3 1v₁ +2v₂ +3v3 4v1 +5v2 +6v3) with respect to the standard bases and show that (-). (1) 5 are both members of the perp of the null space. Prove that (f) is the span of these two. (Hint. See the third item of Exercise 26.) (d) Generalize that to apply to any f: R" → R™.
Transcribed Image Text:✓ 3.27 The material in this subsection allows us to express a geometric relationship that we have not yet seen between the range space and the null space of a linear map. (a) Represent f: R³ → R given by VI V2 V3. with respect to the standard bases and show that (-) 2 is a member of the perp of the null space. Prove that (f) is equal to the span of this vector. → 1v₁ + 2v₂ + 3v3 (b) Generalize that to apply to any f: R" → R. (c) Represent f: R³ → R² V1 V2 V3 1v₁ +2v₂ +3v3 4v1 +5v2 +6v3) with respect to the standard bases and show that (-). (1) 5 are both members of the perp of the null space. Prove that (f) is the span of these two. (Hint. See the third item of Exercise 26.) (d) Generalize that to apply to any f: R" → R™.
✓3.26 Let M, N be subspaces of Rn. The perp operator acts on subspaces; we can ask how it interacts with other such operations. (a) Show that two perps cancel: (M+)+ = M. (b) Prove that MCN implies that N+ C Mt. (c) Show that (M + N) = M²nNt.
Transcribed Image Text:✓3.26 Let M, N be subspaces of Rn. The perp operator acts on subspaces; we can ask how it interacts with other such operations. (a) Show that two perps cancel: (M+)+ = M. (b) Prove that MCN implies that N+ C Mt. (c) Show that (M + N) = M²nNt.
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