Are the following statements true or false?1. If W = Span{X₁, X2, X3} with {X₁, X2, X3 } linearly independent, and if {V₁, V2, V3 } is an orthogonal set in W consisting of non-zero vectors, then {V₁, V2, V3} is an orthogonal basis for W.2. For any scalar c, and vectors u, v E R", we have u · (cv) = c(u · v).3. If x is not in a subspace W, then Xx – projw(x) is zero.4. If y = Z₁ + Z₂, where Z₁ is in a subspace W and z₂ is in W, then Z₁ must be the orthogonal projection of y onto W.5. For a square matrix A, vectors in the column space of A are orthogonal to vectors in the nullspace of A.

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Answered: Are the following statements true or…

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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Let u = (1,3, 5) and v = (-4,3, –7) be vectors in R3. (a) Find u + v. (b) Express u in terms of the standard basis vectors in R3. (c) Find -3u.
Suppose {V1, V2, V3} are linearly independent vectors in a vector space V. Find the number(s) k such that the vectors w₁ = V₁ + kv₂, W2 = V2 + kv3 and w3 = V3+ kv₁ are linearly dependent. Justify your answers. Hint: Assume C₁W₁+C₂W2+C3W3 0 for some scales C1, C2, C3. Express it in terms of lin- ear combinations of V₁, V2, V3. Then, use the linearly independence of {V₁, V2, V3} to get a linear system with k as a parameter and C₁, C2, C3 as unknown variables. {w₁, W2, W3} are linearly dependent if the system only have zero solutions.
Suppose that v = (v1, v2, ..., vn) and û = (u₁, U2, ..., Un) are a pair of n-dimensional vectors. Assume that each component of the vector is a real number, so 7 and u are both members of the set R¹. We will say that and u are "almost the same" when every component of is close to every component of ū. That is, v₁ is close to u₁, v2 is close to u2, etc (practically speaking, "close" means that their absolute difference is small). Assume that we are given the predefined predicate CloseTo(x, y) and the integer constant n. Use them to write a formal definition of the new predicate Almost The Same (7, u) which asserts that n dimensional vector is almost the same as ū. Tip: It is not legal to say i v to refer to a component of v, because is not a set. Instead, use vi to refer to the ith component of v. What set would i belong to in this case?
Let {v1, v2} be an orthogonal set of nonzero vectors, and let c1, c2 be any nonzero scalars. Show that {c\V1, C2V2}is also an orthogonal set. Since orthogonality of a set is defined in terms of pairs of vectors, this shows that if the vectors in an orthogonal set are normalized, the new set will still be orthogonal.
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.
1. Determine if these vectors are linearly independent or linearly dependent. If they are linearly independent show this. If they are linearly dependent, write a nontrivial linear combination (at least some coefficients need to be nonzero) of them being equal to 0. {4.0-6)}
If {v1, v2, . . . , vn} is a linearly dependent set of vectors, prove thatone of these vectors is a linear combination of the other.
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.
1. Let V₁, V2,...,Vn be n vectors in a vector space V. Explain what it means to say that these vectors span V.
Suppose now we have a set of non standard basis vectors v1, v2, .., Un e R" b) which satisfies the following equation: 1 1 WV = I = [v1, v2, ..., vn] where w and vi are the rows and columns of W and V, respectively. Any vector z € R" can be written as a linear combination of the st andard basis vectors v1, v2, ..., Vn as follows: k=1 Determine the coefficients ak in terms of z and wf only.
Suppose V1, V2, V3 is an orthogonal set of vectors in R5. Let w be a vector in Span(V₁, V2, V3) such that V₁ V₁ = 42, V₂ · V₂ = 542, V3 V3 = 36, -126, w - V₂ - 2710, w - V3 w. • V₁ then w= v₁+ = = = -36, V2+ V3.

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