Suppose W is a subspace of R" spanned by n nonzero orthogonal vectors. Explain why W = R". The goal is to show that the subspace W is actually all of R". In general, when are two subspaces known to be the same space? O A. Two subspaces are the same when they are spanned by the same vectors. O B. Two subspaces are the same when they have the same dimension. Oc. Two subspaces are the same when one subspace is a subset of the other subspace. O D. Two subspaces are the same when they are subsets of the same space. To start, assume that W is a subspace of R" spanned by n orthogonal nonzero vectors. Call these vectors v,, V2, V3. ., Vo: By assumption, W= Span (v,, V2, V3, ..., V. Which of the following statements follows immediately from the assumption that the vectors v,, v2, V3, .., V, are orthogonal and nonzero? O A. At least one element of (v1, V2, V3, .., Vn) is zero. O B. R" = Span{V1, V2. V3. .. Vn} Oc. (V1, V2. V3, ..., Vn) is linearly dependent. O D. (V1, V2. V3, ..., V is linearly independent. How does this show that W = R"? O A. The vectors V. V2. V3. ,Vn are in both W and R". O B. The column vectors v,, V2, V3, .., V, span R" if and only if (v1, V2, V3, ., Vn is linearly dependent. O C. The column vectors v1, V2, V3. ., Vn span R" if and only if {V1, V2, V3, ..., Vn is linearly independent. O D. w= (V1, V2, V31 ..., Vn} = R".

Q: Let V = R³ and S be the set of all vectors (x, y, z) in V such that x = r-3s, y = 2r + s an z = s,…

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Q: Which of the following is a subspace of R? O A. All vectors of the form (a, 0,0). O B. All vectors…

A: Subspace of a vector space:  A subset V of ℝn is said to be a subspace of ℝn if 1. The zero vector…

Q: Please go through each option with explanation

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Q: True/False 1) Decide if each of the following statements is true or false. Justify your conclusion…

A: To find: Whether the given statements are true or false.

Q: Let W be the subspace spanned by u₁ and u₂, and write y as the sum of a vector in W and a vector…

A: Firstly we have to find,

Q: 3. Suppose that u and vare nonzero vectors and that S is a subspace. Prove that if u ES and v E St,…

A: Let u and v be non-zero vectors and S be a subspace. Assume that u∈S and v∈S⊥. This means that…

Q: c) Mark each statement True or False. Justify the answer. A subspace is also a vector space. • R? is…

A: Here , A subspace is olso a vector space: it is true.  R^2 is a subspace of R^3 : it is false.

Q: V₁ 1₁3 G a. Is w in (₁,₂,3)? How many vectors are in (V₁, V₂, Va)? b. How many vectors are in…

A: Three vectors v1,v2,v3 are given. (a) First one needs to check whether another vector w is in the…

Q: Let W { (а, b, а + 2b + с, 2а — Зc), а, ь, с€ R} %3D - 1. Show that W is a subspace of R4. 2.…

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Q: Suppose W is a subspace of R" spanned by n nonzero orthogonal vectors. Explain why W =R". The goal…

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Q: 3. Let A be the vector (6, 7, 0, 9, 15), B be the vector (a, b, c, d, e), and C be the vector (-4,…

A: Answer:- Given, the vectors A  = (6, 7, 0, 9, 15), B = (a, b, c, d, e) and  C = (-4, -1, -2, 0, -15)…

Q: Determine whether each of the following statements is true or false. You do NOT need to explain. Let…

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Q: Find all values of h such that vector y will be in the subspace of R spanned by vectors 3. V1, V2,…

A: we will first find the equivalent matrix and then reduce it to row echelon form

Q: Let W be the subspace spanned by u₁ and u₂, and write y as the sum of a vector in W and a vector…

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Q: 13. Find a basis for the subspace spanned by the vectors What is the dimension of the subspace? C100

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Q: Determine whether the set W is a subspace of R³ with the standard operations w = {(a, -a,0): a is a…

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Q: Let V = R³ and S be the set of all vectors (x, y, z) in V such that x = r-3s, y = 2r+s and z = s,…

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Q: a2 E R³ are subspaces of R³? (No justification necessary.) Which of the following sets of vectors аз…

A: Vector subspace : A subset W of a vector space V  over the scalar field K is a subspace of V if and…

Q: Which of the following is a subspace of R? O A. All vectors of the form (a, 0, 0). B. All vectors of…

A: To choose correct option we will use the following theorem Theorem: A non-empty subset W of a vector…

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A: To find wheather A spans R3 we have to check if it forms a basis in R3

Q: a b a b M22 space of a, b, c e R and K= 20 - Which of the sets is a subspace under standard…

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Q: Let W be the subspace spanned by u, and u₂, and write y as the sum of a vector in W and a vector…

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.