Suppose {u1,u2,...,un} is a set of vectors that span V, and that {v1,v2,...,vk} is an independent set of vectors in V. Explain why v1=x1u1+x2u2+...+xnun must have a nontrivial solution. Explain why we may assume x1≠0. (Hint: x1 is the first unknown because u1 was the first vector in our spanning set.) Explain why this means u1 is in the span of {v1,u2,...,un} Explain why this means {v1,u2,...,un} also spans V.

Q: 8. If {V1, V2,..., ,}is a linearly dependent set of vectors, prove that one of these vectors is a…

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Q: 4. Express PQ-RQ+SR-PS+RS as a single vector.

A: Expression

Q: Draw vector 1/2CD - 2FE

A:

Q: Let V₁, V2, V3 be linearly independent vectors in R". Show that {v₁ + V₂, V₂ + V3, V3 + ₁} is also…

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Q: If possible, write the vector w as a linear combination of the vectors V₁, V2, ..., Vk. If it is not…

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Q: 9 One requirement can be met while the other fails. Show this by finding (a) A set of vectors in R²…

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Q: Suppose that {x1, x2, x3} is a linearly independent set of vectors in a vector space V . Must {x1,…

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Q: Three unique non-zero vectors a, b, and C are defined by four points in space. Select all of the…

A: Proceed as shown below

Q: If p = a₁ + a₁x + a₂x² and q = bo + b₁x + b₂x² are any 2 vectors in P². Defined function: f(p, q) ==…

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Q: Let y = and u = Write y as the sum of two orthogonal vectors, y, in the span of u, and y, which is…

A: Solution

Q: 5. i ',, Find the solution space and write it as a linear combination of vectors. - x3 + 5x4 X2 +…

A: Given that x1-x3+5x4=0 x2+4x3-3x4=0

Q: 5. Determine which sets of vectors are orthogonal. (a) 1 ↑ 1 0 -5 2 1 5 -309 (b) -4 2

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Q: What if the vector v was [ 7, 11, 3]

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Q: Given the vector (x <- c(5, 2, 7), what would x^2 return? O [1] 78 O [1] 196 O [1] 25 4 49 O [1] 10…

A: Given that, the vector is        ,we need to find .

Q: (In set notation: (| x ER).) In the following, determine whether the statements are True or False,…

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Q: Suppose ₁, 2, 3 is an orthogonal set of vectors in R³. Let w be a vector in Span(V₁, V2, V3) such…

A: Given: v1, v2, v3 is an orthogonal set of vectors in ℝ5.           v1·v1=30, v2·v2=25.25, v3·v3=9…

Q: 2 aj = a2 = az = b = Determine if vector b is a linear combination of vectors a1, a2, and аз. Show…

A:

Q: 3) For each set of vectors below, determine if the set is dependent or in- dependent. If the set is…

A: We will first check the linearly independence of this set of three vectors

Q: 6. If If vectors (5,2,-1) × (2,-1, a) = (5,-12,-9) then determine the value of a.

A: Given: 5,2,-1×2,-1,a=5,-12,-9

Q: 2. a. In each part express the vector as a linear combination of 2+x+4x², p₂ = 1 − x + 3x², and p3 =…

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Q: Suppose that u₁ and u2 are vectors with ||u₁|| = 2 and ||u2|| = 3. You are also given that u₁ · U₂ =…

A: Given that  and  are vectors such that .Given that  and  for .We need to find  for .We need to…

Q: Find the vector determined by 18 || = B = {[.]. 9 6 12 3 9 2 -8 " where B = 3 [-]

A:

Q: If vector u=2:1+2:/ and vector V =5./+4/, simplify each of the following, writing your final answer…

A: Given:    u = 2i + 2j    v = 5i + 4j

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.