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(d) Using your result from part (c), write the vector y = and a vector in W¹. 6 as the sum of a vector in W

(d) Using your result from part (c), write the vector y = and a vector in W¹. 6 as the sum of a vector in W

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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↑ D (a) note 0₁.0₂ (b) So fui, ū₂ ) TS = define v₂ = :.U₂ (Ⓒ) Note || Step 3 11 TE Now 20 Step 4 1 2 So = proj, (2₁) = P= and by Gram-Schindt Orthogonalization process that proj (ei) W powo) (ez) proj 11 = = Vq ין that 11 110q||| ( = = NO = 3 V. -213 Proj (2₂) = = 1/3 O W 213 -2/3 1/3 2/9 -4 <ej.u₁> U₁ + <e;• U₂> U₂ 8/9 -2/9 2/9 0. ฟง 219 none that {ei) i21,2,314 9 819 8/9 -219 219 0 4x2 + 2x0 +0x1 N/O या <V₂_U₁) U₁ 2 0 213 - U₁ 8 -2 -213 1/3 0 213 > -213 (DOX 213 ban's of W. t 0 -219 1/9 O 219 (4.2.0.4) 16+4+0+16 ²/3, 0, 7 ) ² 2 -10 11 2 1 0 2 an orthonormal 4+0+1+4 213 O 13 +0 + No-d + 2 243 and proj₁ (24) = <BP ,,ea> Uit <U₂, e4$ Uz -213 1/3 O 213 basis elements 2/9 0 U₂ 2 +3) g 1/9 219 O • (U₂) +4x2 = ( 0 219 219 2 2 8 ²7-1 T 8/9 V₂ 13 213 O 2/3 are sta No-№ T -213 43 213 0 43 213
Transcribed Image Text:↑ D (a) note 0₁.0₂ (b) So fui, ū₂ ) TS = define v₂ = :.U₂ (Ⓒ) Note || Step 3 11 TE Now 20 Step 4 1 2 So = proj, (2₁) = P= and by Gram-Schindt Orthogonalization process that proj (ei) W powo) (ez) proj 11 = = Vq ין that 11 110q||| ( = = NO = 3 V. -213 Proj (2₂) = = 1/3 O W 213 -2/3 1/3 2/9 -4 <ej.u₁> U₁ + <e;• U₂> U₂ 8/9 -2/9 2/9 0. ฟง 219 none that {ei) i21,2,314 9 819 8/9 -219 219 0 4x2 + 2x0 +0x1 N/O या <V₂_U₁) U₁ 2 0 213 - U₁ 8 -2 -213 1/3 0 213 > -213 (DOX 213 ban's of W. t 0 -219 1/9 O 219 (4.2.0.4) 16+4+0+16 ²/3, 0, 7 ) ² 2 -10 11 2 1 0 2 an orthonormal 4+0+1+4 213 O 13 +0 + No-d + 2 243 and proj₁ (24) = <BP ,,ea> Uit <U₂, e4$ Uz -213 1/3 O 213 basis elements 2/9 0 U₂ 2 +3) g 1/9 219 O • (U₂) +4x2 = ( 0 219 219 2 2 8 ²7-1 T 8/9 V₂ 13 213 O 2/3 are sta No-№ T -213 43 213 0 43 213
1. Let W = Span{V₁, V₂} with v₁ = 2 0 and 72 = and a vector in W¹. 0 1 (a) Show that {7₁, 7₂} is an orthogonal basis for W. (b) Construct an orthonormal basis {₁, ₂} for W. (c) Construct the projection matrix P = UUT such that projw ý = Pỹ. -6 8- 9 6 (d) Using your result from part (c), write the vector y = as the sum of a vector in W
Transcribed Image Text:1. Let W = Span{V₁, V₂} with v₁ = 2 0 and 72 = and a vector in W¹. 0 1 (a) Show that {7₁, 7₂} is an orthogonal basis for W. (b) Construct an orthonormal basis {₁, ₂} for W. (c) Construct the projection matrix P = UUT such that projw ý = Pỹ. -6 8- 9 6 (d) Using your result from part (c), write the vector y = as the sum of a vector in W
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