'S' is a collection of vectors. Show that the rank of S is equal to the dimension of Span(S). Please show step by step in detail, I have a hard time with these abstract questions.

Q: When the vector b = -2 is projected onto the line spanned by the vector a = 1 what is the error 2…

A: The projection of b onto a line spanned by a vector a is  b.a^=b.a|a| Given:  b=[-2-2-1]…

Q: In each part, find all the roots and sketch them as vectors in the complex plane. (a) (-i)//² (b)…

A: Note : Since you have posted a question with multiple subparts, we will provide the solution only to…

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: 3.1 Suppose that u and v are non-parallel, non-zero vectors in R². You will show that c-span{u, v}…

A: Given: In this problem, the given details are, Here, u and v are non-parallel and non-zero vectors…

Q: (1) Find the area of the triangle and indicate its type.

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Q: Consider a parallelogram generated by the vectors (1, y, -2) and (3, 0, 1). For what value(s) of y…

A: If   and are adjacent sides of a parallelogram then its area is

Q: Find the vector equation for the line of intersection of the planes 3x - 4y + 3z = -1 and 3x + 3z =…

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Q: Maximize: P = 22x + 15y Subject to: 3x + 4y 0

A: “Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 5. (a) Given A = a,5-ay2 + a₂, find the expression of a unit vector of B such that B/A and BL A, if…

A: Solution 5(a): Given that A→=5ax-2ay+az and  B→∥A→ if B→ lies in the xy-plane. So, B→=5bx-2by. Unit…

Q: What is the least number of linearly independent vectors needed to span a line? [Enter a whole…

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Q: 3 -3 2 6 -3 7 (1.8) Consider the set spanned by the columns of B = 6 3 16 9 -3 21| Please write an…

A: Here,B=3-326-3763169-321∴By reducing B into echelon form by applying row operation, we…

Q: Let L₁ be the line passing through the points Q₁=(4, -2,-4) and Q₂=(3, 1, -7) and let L₂ be the line…

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Q: 9 Find the distance from the point (-7,-3,4) to the xy-plane.

A: Distance of a point (a,b,c) from the plane lx+my+nz=p is (la+mb+nc-p)/sqrt(l^2+m^2+n^2)

Q: 39. For each condition, find unit vectors å and b or explain why no such vectors exist. (a) |a+헤-0…

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Q: The sum of the squares of the direction cosines of a vector in space add up to ..............

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Q: I have the following vectors, I have a problem with understanding why one becomes 'a+b' and the…

A: Vectors are geometric objects with both magnitude and direction. We will consider two vectors, a and…

Q: (b) Find the points in which the required line in part (a) intersects the coordinate planes.

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Q: write the general solution to  x+y+z+w=1 x+y-z-w=1 as a vector in the form of y or w

A: NOTE: Refresh your page if you can't see any equations. . the system Ax=b is

Q: (b) Let v₁ = (-2, 1,5), v₂ = (0, 1,4), V3 = (-2,4, 17). What is span(V1, V2, V3)? Could it be all of…

A: Given vectors are v1=(-2,2,5), v2=(0,1,4),v3=(-2,4,17) we have to find span(v1,v2,v3)

Q: use the vectors u = , v = , and w = a) Find which pair(s) of vectors from that set is/are…

A: We have to check

Q: The distance from (-3,-4,-7) to the yz-plane is (A) -4 (B) 4 (C) -3 (D) 3 (E) 7

A: The distance(d) between two points (x1,y1,z1) and (x2,y2,z2) in a cartesian coordinate system is…

Q: What can you say about the relative position of two nonzero vectors if their dot product is zero?

A: Dot product of two non-zero vectors is zero. We know that two vectors are perpendicular if and only…

Q: ? (2,3), (0,0) }

A: Let S=2, 3, 0, 0. We need to find whether the set S spans a line or all of ℝ2.

Q: 4. Find two vectors and such that (i) both and are perpendicular to = (1,0,-1) and (ii) is…

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Q: If a = (4, 0, -1), find a vector b such that comp₂b = 3. b = X

A: a=<4,0,-1>ab=3ab=a→.b→a→let b=<b1,b2,b3>