24. Let S be a basis for an n-dimensional vector space V. Show that if V1, V2, ..., v, form alinearly independent set of vectors in V, then the coordinate vectors (v)s. (V2)s..... (v,)sform a linearly independent set in R", and conversely.25. Using the notation from Exercise 24, show that if v, v2, ... v, span V. then the coordinatevectors (v,)s. (v2)s.. (v,)s span R", and conversely.

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Answered: 24. Let S be a basis for an…

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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5. Find the orthogonal projection of the vector b = vectors V₁ = -1- and v2 = onto the subpace E spanned by the (Note that v₁ V₂).
4. Let S = {v,,v2, V3, V4, V5} where, 3 1 2 3 3 ,V2 9. ,V3 2 la -1 4 Find a basis for the space span(S) which is a subset of S. Then express each vector that is not in the basis as a linear combination of the basis vectors.
8.) find a vector of norm 2 that is orthogonal to the three vectors, u=(2,1,0), v=(-1,- 1,0) and w=(4,0,0) 9.) find scalars a, b and c such that a(2,0,1) +b (1,3,2)+c(1,0,1)=(7,3,4)
16, w be a vector in Span(v₁, V2, V3) such that = 11, V₂ V2 V1 V1 w v₁ = -55, w v₂ then w = • Suppose V1, V2, V3 is an orthogonal set of vectors in R5. Let . = . . = 46, V3 V3 184, w = v1+ . = V3 = 64, V2+ V3.
7. Find the coordinates of the vector v relative to the ordered basis B. a) B = {x² + 2x + 2, 2x + 3,−x² + x + 1} v=-3x² + 6x + 8 14 D b) B = {[1₁¹] [ = (₁ 3²1 -CAC) = B₂ -QAED = 0 v= c) B₁ v= 22
3. Prove the following to establish theorem 7. (a) If the set of vectors {₁, 2, 3, ...} in R" are linearly dependent then at least one of the vectors can be expressed as a linear combination of the others. (b) If at least one of the vectors from the set containing vectors ₁, 2, 3, ... in Rn can be expressed as a linear combination of the others then the vectors form a linearly dependent set.
2. In a vector space V with inner product (,), the vectors u and v are orthogonal and ||u|| ||v|| = 5. Determine the value of (u + 3v, 2u - 4v). = 3 and
1- Determine whether the following sets of vectors are linearly dependent or linearly independent. a) (-3,0,4), (5,-1,2), (1,1,3) in R³ b) (1,0,0), (2,2,0), (3,3,3) in R³ 2- Verify that the set of vectors (-0.6, 0.8, 0), (0.8, 0.6, 0), (0, 0, 1) form an orthonormal basis for R³, then obtain an orthonormal basis from them. Express (3,7,-4) as a linear combination of the orthonormal basis. 3- Determine whether the following sets of vectors are linearly dependent or linearly independent. a) (1,1,0), (0,0,1), (0,1,1) in R³ b) (1,0,0), (2,2,0), (3,3,3) in R³ 4- Verify that the set of vectors (-0.6, 0.8, 0), (0.8, 0.6, 0), (0, 0, 1) form an orthogonal basis for R³, then obtain an orthonormal basis from them. Express (2,3,-4) as a linear combination of the orthonormal basis. 5- Find the coordinates of the vector u = (0,25, 1) with respect to the orthogonal basis v₁ = ( 3,4,0), V₂ = (−4, 3, 0 ) and v₂ = (−4,3,0) 6- Find the coordinates of the vector U = (7,5,2) with respect to the basis V₁ =…
Suppose V1, V2, V3 is an orthogonal set of vectors in R5. Let w be a vector in Span(V1, V2, V3) such that V1V1= 27, V2 V2 = 54, V3 V3 = 16, . . w.v₁ = -27, w v2 = 378, w V3: ₁+₂+ V3. then w = = 32, =

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24. Let S be a basis for an n-dimensional vector space V. Show that if v, v2. ...,v, form a linearly independent set of vectors in v, then the coordinate vectors (v)s. (V2)s. .... (v,)s form a linearly independent set in R", and conversely.