A sum of scalar multiples of two or more vectors (such as c,u + c,v + c,w, where c, are scalars) is called a linear combination of the vectors. Let i = (1,0), j = (0, 1), u = (1, 1), and v = (-1,1). For arbitrary real numbers a and b, expre ss (a, b) as a linear combination of u and v.

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

A:

Q: If three vectors vị, v2, V3 E R° are linearly dependent, then one of them must be a scalar multiple…

A: GIVEN THREE VECTORS v1, v2, v3 belongs to R^3 are linearly dependent

Q: If i = (a, b, 0), ï = (a, 1, 6), and (1,1, 12) are vectors in R Determine the value of b so that and…

A:

Q: 5. Determine the number of vectors (x1,...,x„), such that each x; is either 0 or 1 and i=1

A: (x1,x2,...,xn) is a vector with xi is either 1 or 0.  Now, ∑i=1nxi ≥ k implies at least k of the…

Q: u = (2, 3,-1) v= (0, 1, -2) w = (2, 4, -3).

A:

Q: 7. "Let &, n, ở be real numbers and consider vectors (1,5,8²); (1,n,n²); (1,0,0²). Determine the…

A:

Q: Show that the vectors V1 = ( 1, 1, 2, 4 ), V2 = ( 2, -1, -5, 2 ), V3 = ( 1, -1, -4, 0 ) and V4 = (…

A:

Q: let v = (1, 3, −2) and w = (2, −1, 4). Find all the vectors orthogonal to both v and w.

A: Given: The vectors v=1,3,-2 and w=2,-1,4. To find: The vectors orthogonal to both v and w.

Q: v, =,v, =, v, =

A: To add or subtract two vectors, add or subtract the corresponding components of each vector. For…

Q: The vectors --[+]»-[i]»-[3]m² -]»-[3]»=[] V2 = span R2 but do not form a basis. Find two different…

A:

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: Show that there are no vectors u and v such that 11u 11= 1, ||v|| = 2, and uxv= 3.

A: If u and v are two vectors, and the angle between these two vectors is θ Then the magnitude of the…

Q: Prove that the set of vectors B = {(1,2,3), (3,2,1),(0,0,1)} is linearly independent and spans R³.

A:

Q: Let u, v, and w be linearly independent vectors. Show that (u+v), (u-v), and (u-2v+w) vectors are…

A:

Q: Let x=⟨1,1,1⟩ and y=⟨0,3,−2⟩.Express y as the sum of two vectors: one parallel to x, the other…

A: Given that: x=(1,1,1) and y=(0,3, -2) are two  vectors. Let us say,  w = (a, b, c ) is a vector…

Q: Show that the vectors v, = (0, 3, 1, -1); v =(6, 0, 5, 1); v (4, -7, 1, 3) form a linearly dependent…

A: Given vectors v1 = (0, 3, 1, -1); v2 = (6, 0, 5, 1); v3 =(4, -7, 1, 3) . Let c1. v1 + c2. v2 + c3.…

Q: If {v1, v2, . . . , vn} is a linearly dependent set of vectors, prove thatone of these vectors is a…

A:

Q: 1. Determine whether the vectors vi (1,2,3), v2 (2,4, 5), and v3 (1,0,0) are independent.

A:

Q: Show that the vectors 2 -2 2 uj = 2 uz = 3 uz = 7 1 -5 -3 are linearly dependent. Find all r1, x2,…

A:

Q: Find all values of b for which the vectors are orthogonal 〈b, 3, 2〉,〈1, b, 1〉

A: Concept - If two vectors say u and v are orthogonal, then their dot product is equals to zero.That…

Q: Prove that if the vectors a,b parallel, then āxb=0. are %3|

A:

Q: Determine whether the given vectors (1, 1,-1, 1), (1,-1, 2,-1), (3, 1, 0, 1) in R* are lincarly…

A: Given vectors are: (1, 1, -1, 1) , (1, -1, 2, -1) , (3, 1, 0, 1) We have to determine whether the…

Q: If it is known that the internal product of the vectors ā (4, x, -2) and ƃ (1, x + 1,5) is 6,…

A:

Q: Consider the vectors in R2 3 2 (3) Express as a linear combination of the vectors Select one:…

A: Here we have to find the linear combination.

Q: 8.) find a vector of norm 2 that is orthogonal to the three vectors, u=(2,1,0), v=(-1,- 1,0) and…

A: Just assume a vector and apply criteria of perpendicular of two vectors ..as I have done solution in…

Q: (a) Show that the vectors v1 = (1, 2, 3, 4), v2 = (0, 1, 0, – 1), and v3 = (1, 3, 3, 3), form a…

A: (a) The given vectors are: v1=(1,2,3,4), v2=(0,1,0,-1), v3=(1,3,3,3)

Q: . Show that the vectors V1=(0, 3, 1, -1); V2=(6, 0, 5, 1); V3=(4, -7, 1, 3) form a linearly…

A:

Q: ) Show that the three vectors vị = (1, 2, 3, 4), v2= (0, 1, 0, – 1), and v3 = (1, 3, 3, 3) form a…

A:

Q: When V is spanned by {v1, v2, . . . , vk} and one of these vectors can be written as a linear…

A:

Q: Show that the column vectors (1,1,2,4),(2,-1,-5.2) (1,-1,-4,0) and (2,1,1,6) vectors in R are…

A:

Q: If the vectors a and b are such that a = kb where k is a positive scalar, then the vectors a and b…

A: Given statement: If the vectors a and b are such that a=kb where k is apositive scalar, then the…

Q: 20. Find two vectors of norm 1 that are orthogonal to the vectors u = (2,1,-4,0), v = (-1,–1,2,2),…

A: To find two vectors of norm 1 that are orthogonal to the below vectors.…

Q: Determine if the vector 2 b = 3 5 can be written as a linear combination of the vectors V₁ = 1 and…

A:

Q: 19. Find two vectors of norm 1 that are orthogonal to the vectors u = (2,1, –4,0), v = (-1,–1,2,2),…

A: Step:- 1 Given that u=2, 1, -4, 0,  v=-1, -1, 2, 2,  w=3, 2, 5, 4 Now, first we check the given…

Q: Suppose a and b are vectors such that axb=(1,

A: Given, a→ and b→ are vectors such that a→× b→=(1, 0, 2).

Q: Show that the column vectors (1,1.2,4), (2,-1,-5,2). (1,-1,-4,0and (2,1,1.6) vectors in R are…

A: We have to prove that the given Column vectors are linearly dependent in R4.

Q: Determine k so that the vectors (1,–1, k - 1), (2, k,-4) and (0, 2 +k,-8) in R linearly independent.…

A: The detailed answer and explanation is given below:-

Q: Express the vector x as a linear combination of u,, u2, and uz. – 2 3 - 2 ,x = 0 u2 = 5 u3 = – 4 u1…

A: Given u1=-201, u2=356, u3=-26-4 and x=-99-38

Q: What must be a if the vectors (1,-a,1,1). (-1,a+2,–2, a –1), (0,2, –1,a² +a-4) are linearly…

A: We have to solve given problem:

Q: Find all vectors in R2 that are orthogonal to u = (2,1).

A:

Q: 7. Find the cross product of the vectors (a) u=(1,2,-3) v=(-4,1,2) (b) u = (0,1,- 2) v = (3,0, - 4)

A:

Q: Find two vectors of norm 1 that are orthogonal to the vectors u= (2, 1, - 4, 0), v= (-1, - 1, 2, 2)…

A: Let X=x,y,z,t be the vector that is orthogonal to u= (2, 1, - 4, 0), v= (-1, - 1, 2, 2) and w= (3,…

Q: A sum of scalar multiples of two or more vectors (such as c,u + c,v + c,w, where c, are scalars) is…

A: Given i=1,0 and j=0,1

Q: Let a = (4, 8, -6) and b = orthogonal to a. (2, 2, –7) be vectors. Decompose the vector b into a…

A: We are given a=(4,8,-6) and b=(2,2,-7) be the two vectors. We have to decompose the vector b into a…

Q: Let x, y, and z be vectors in Cn and let α and β becomplex scalars. Show thatz, αx + βy} = α z, x} +…

A: Given: z,αx+βy=αz,x+βz,y

Q: A2. Express v = (1,-2,5) in R as a linear combination of the vectors u; (1, 1, 1), %3D uz = (1,2, 3)…

A:

Q: Determine if the vector v is a linear combination of the remaining vectors.

A: Given vectors are : v =3.22.0-2.6,  u1 = 1.00.44.8 ,  u2 = 3.41.4-6.4 , u3 = -1.20.2-1.0 If the…

Q: 5. Show that if vị and vz are linearly independent vectors in R?, then any vector w E R² can be…

A:

Q: Prove that two vectors are linearly dependent (taken as a set of vectors) if and only if one vector…

A: Let X and Y be two vectors. Suppose these two vectors are linearly dependent. [To prove: One vector…