Determine whether the given set S of vectors is closed under addition and closed under scalar mul-tiplication. In each case, take the set of scalars to be the set of all real numbers. The set S := {A ∈ Mn(R) : A is upper or lower triangular}.  R=all real numbers

Q: 1. Determine if each set is orthogonal, orthonormal, or neither. If it orthogonal, normalize the…

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Q: Determine whether the vectors u = 2 – x + 3x?, v = 4 +x + 2x?, and w = 8 - x + 8x² span P2 (the set…

A: The set {p1,p2,p3} will span p2 if and only if they are independent  so, here we test if the vectors…

Q: The column space of A is the set of all vectors that are linear combinations of the columns of A. 3…

A: Given matrix A=24-21-2-57337-86  and  u=3-2-10  and  v=3-13 Remarks If AX=0then x in Null(A)

Q: Show that the set is linearly dependent by finding a nontrivial linear combination of vectors in the…

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Q: Show that if S = {v1,v2,.., Vn} is an orthogonal set of vectors, then for any scalars a1, a2, ... ,…

A: If S is an orthogonal set of vectors then to deduce that for any scalars set T is also an orthogonal…

Q: 9. Find all real values of c such that the vectors and are linearly independent.

A: Given matrices are 2c0 and cc+30   We know the formula or the quadratic equation ax2+bx+c=0 is…

Q: Let S be a set of m vectors in R" with m > n. Select the best statement. O A. The set S could be…

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Q: Determine whether b is in the column space of A, and if so, express b as a linear combination of the…

A: Given a matrix A=356-10 and a vector b = -216. Express b as a linear combination of the column…

Q: Determine whether each statement is true or false. If a statement is true, give a reason or cite an…

A: (a) True

Q: Find the Euclidean norm of the given vector. 10 v = 24 Hint: 102 = 100, 242 = 576 ||v|| = Ex: 1

A: Given vector is: v=1024

Q: Find the maximum and minimum values, and a vector where each occurs, of the quadratic form subject…

A: According to the given information, it is required to find maximum and minimum values and a vector…

Q: find all vectors v⃗  in 2 dimensions having ||v⃗ ||=41 where the j⃗ -component of v⃗  is 9j⃗ .

A: Our objective is to find all the vectors v having ||v||=41 and the j-th component 9j.

Q: Determine whether the set of vectors in P, is linearly independent or linearly dependent. S = {1 +…

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Q: Let S be a set of m vectors in R", where m > n. Select the best statement below. The set S could be…

A: Here, we need to select the best statement.

Q: Which of the following vectors MUST be linearly dependent? here a,b,c,d,e,f,g,h,i,j,k & l are real…

A: The objective is to find that which of the vector must be linearly dependent It is given that…

Q: Find the value of k for which the vectors 4 and 4 k are orthogonal. k =

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Q: the vectorS form an orthog Express x as a linear combination of the u's. u, + U3 (Use integers or…

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Q: Find the value of k for which the vectors -4] -5 and -5 k are orthogonal. k =

A: Let the two vectors be- a →=-20-51        and  b→= -4-50k It is given that both the vectors are…

Q: Find a value of k such that the given vectors are linearly dependent [ -2 4+ k 4 ,X2 = 2 X1= ,X3 =…

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Q: (b) 4x Find any values of x such that the vectors 3 and are parallel X-1 to each other.

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Q: (a) Find the set of all solutions to Ax = 0.(b) Express each solution as a linear combination of…

A: Given matrix is: We have to find the set of all solutions of AX = 0. Let X = [x   y   z] Then we…

Q: Find the norm of and the unit vector u in the direction of if 3 4 1-2) 18 |||| 13 = - 18 ||

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Q: Could a set of three vectors in R* span all of R4? Explain. What about n vectors in RM when n is…

A: Determine whether a set of three vectors in R4 span all of R4 as follows. Consider a matrix A of the…

Q: Determine the real number a such that u X v| and i are orthogonal, where u = 2i + 4j + 3k and v = 8i…

A: u=2i+4j+3kv=8i-10j+αk

Q: Find the value of k for which the vectors and 4 -4 are orthogonal. k =

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Q: a Let V be the set of vectors of the form b,where a, b, c are real numbers, then V is not closed…

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Q: 79. Show that if a, b are nonzero vectors such that a 1 b, then there exists a vector X such that a…

A: To show a×X=b if a, b are nonzero vectors and perpendicular to each other.

Q: Given the vectors V1 = (x, 1,1), V2 = (1, x, 1) and V3 = (1,1, x). S = { V1, V2, V3} of the set Find…

A: 3 vectors form a basis of R3, if they are linearly Independent. Vectors are linearly Independent if…

Q: Show that the set is linearly dependent by finding a nontrivial linear combination of vectors in the…

A: First we will express vector s1 a linear combination of other vectors s2 and s3soing so will give us…

Q: a) Find all the real numbers à such the vectors 1- x 1+x and x+1 1 are orthogonal.

A: Two vectors are said to be orthogonal if their dot product is zero. Dot Product There are two…

Q: Q1. Construct an orthogonal set of vectors from the following vectors: X1 = k,X2 ,X3 %3D 14]

A: Use the usual inner product in the vectors and Gramm-smidth orthogonalisation process.

Q: Show that the set is linearly dependent by finding a nontrivial linear combination of vectors in the…

A: Solution:      If k=1 then (0,0)=-47(3,4)+167(-1,1)+1(4,0)

Q: Suppose v1,..., Vk are linearly independent vectors in R" (1 <k < n). Then the set Xk =…

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Q: 150. Determine the real number a such that vectors a = -3i+2j and b = 2i+aj are orthogonal.

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Q: 149. Determine the real number a such that vectors a = 2i +3j and b = 9i+aj are orthogonal.

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Q: 5. For which real values of à do the following vectors form a linearly independent set in R³?

A: Solution; Consider the given vectors are: v1→ = λ, -12, -12v2→ = -12, λ, -12v3→ = -12, -12, λ Since…

Q: 9. Find all real values of c such that the vectors and +3 are linearly independent.

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Q: Find an explicit description of Nul A by listing vectors that span the null space. 120-30 A 0 0 1-4…

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Q: Show that the set is linearly dependent by finding a nontrivial Ilinear combination of vectors in…

A: Given S = {(3,2), (-1,1), (2,0)}

Q: 2. Determine all real numbers x e R for which the vectors (x, 0, 1), (2, x, 3), and (4, 5, 6) ar…

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Q: If a,b,c are non-zero vectors such that a.b = a.c, then find the geometrical relation between the…

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Q: Determine whether the set of vectors in P, is linearly independent or linearly dependent. S = (-x…

A: To verify the given set of vectors are linearly dependent or independent. Let…

Q: 6. Let ū, ū, and w be vectors. Explain why any linear combination of ū and ở is also a linear…

A: Consider any two vectors x→ and y→, let the linear combination of these vectors be denoted by a→,…

Q: Let a, b, and c be n-vectors and let k be a real number.(a) Show that a · b = b · a.(b) Show that (a…

A: Given, Let a, b, and c be n-vectors and let k be a real number.(a) Show that a · b = b · a.(b) Show…

Q: If Q: R" → R is a quadratic form, explain how to determine the maximum and minimum values that Q…

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Q: 1. Consider the vectors x¡ = (8, 6)' and X2 = (4, –1)' in R?. %3D (a) Determine the length of each…

A: (a)

Q: (b) In each part, determine whether the given vector 2x - x2 +x + 3 € P3 (R) is in the span of S =…

A: i) 2x3-x2+x+3∈P3ℝ is in the span of S=x3+x2+x+1, x2+x+1, x+1 if…

Q: Three of the vectors of a vector space of quadratic polynomials p1 (x) = x2 – 1, p2(x) = a² + x,…

A: Solve the following problem

Q: Assume we have 4 vectors v1, v2, V3, V4 from R" and that we can write every vector b in R" as b =…

A: Since we can write any vector of Rn in the form of 4 vectors v1,v2,v3,v4.