9. Find a basis for the hyperplane a where a-(2, 1. -3).

Q: 2) Show that S ER° defined by S = {(1,0, 5), (0, 2, -1)} is linearly independent. Extend S to be a…

A:

Q: Let a = (-3, –4, –4) and b = (2, 2, 4) . Find scalars s and t so that sa + tb = (20, 4, 32) %3D %3D

A: We have to find the values of scalers s and t for a=-3,-4,-4 and b=2,2,4.

Q: 2, Xz} is a basis of V, (R). show that {, +X, X2 + X3, X3 +Xi} is also a basis of V, (R),

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Q: Find a basis for the plane 4 + 3y + 5z = 0 in R. %3D

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Q: Compute the orthogonal projection of v= 7 onto the line through 4 2 and the origin.

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Q: Apply the Gram-Schmidt process to find an orthogonal basis for S. 3 S = span 4

A: In this question, we find the orthogonal set of basis by the Gram-Schmidt process.

Q: 4. Apply the principal axis transformation to the curve of second order -x2 + 6xy – y? – 22.x+ 18y –…

A:

Q: Consider the line L spanned by {(1, 2, 3)}in R. Find a basis of L-. a) {(0, –3, –2)} b) {(1,0,0),…

A: L is a line spanned by a single vectors  in R³, so perpendicular of L has dimension 2.

Q: Find out if the given system of vectors is a basis of the four-dimensional Euclidean space R*: 2

A:

Q: Is this the basis? What is the rank of A?

A: According to the given information, it is required to find the basis and rank of matrix A. The…

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

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Q: Q2. Leb u, and v be two livearly ondependent vecetors in R? Do thiy necussavily form a basis of R?

A: We have to write about vector u and v

Q: 13. Do the set of vectors S = { x²+2x-4, 6x² +7x+3, 2x²+x) a basis for P2? Refer to the definition…

A: The given set is;  S=(x2+2x-4, 6x2+7x+3,2x2+x).

Q: 5.3. Prove that a linearly independent system of vectors v1, V2, . .. , Vn in a vector space V is a…

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Q: et a1, a2 and a3 be colums of A, i.e A = [a1,a2,a3] and let W = span {a1,a2}   Find an orthonormal…

A: Given A=134203166 To find an orthonormal basis of W and distance between a3 and W.

Q: Find a basis for the set of vectors in ℝ2 on the line y = 5x.

A:

Q: Show that the three points (x1, y1), (x2, y2), and (x3, y3) in a plane are collinear if and only if…

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Q: 1 Compute the orthogonal projection of v = onto the line L through -6 and the origin. 2 projz (7) 2.

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Q: 個- to Ez in R². 3 a) Compute the change of basis matrix from 8 %3D

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Q: 4. Present the standard basis vectors e, and ez in R? as linear combinations of (3, 1) and (1, 3).

A:

Q: 14. What are the conditions on a set of vectors S to be a basis for a vector space V?

A: If V is any vector space and  S = {v1, v2, ...... Vn} is a set of vectors in V, then the set S is a…

Q: 14 - Which one is a basis for the vector space R3 ? 1

A: We have to check

Q: Find a basis fo

A: Given,            A=3-916-26-161-38

Q: 2 Find an orthogonal basis for the space spanned by 4 and

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Q: 3) Let S contain vectors of the form f(x) = a + bx + cx² + dx³ : f(-2) = 0 and f'(-4)= 0: Then a…

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Q: Compute the orthogonal projection of = 3 onto the line L through -2 and the origin 2 proj, (v) =

A:

Q: Suppose {u, } form an orthogonal basis in R2 and y = 2ū - 7 where u = (8) 6 If W = Span{}, what is…

A:

Q: 2. Show that the set S = {P1, P2, P3} is a basis for P2 (space of polynomials of degree less or…

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Q: Take L: V --> W, from P2 --> P2 and L(p) = p + p', with basis B = {1 + x + x2, 1 + x, 2x + x2} and…

A:

Q: Assume that: U1= -1 -1 u2= 1 And W=span {U1, U2}. -Find a basis to W-.

A: Here   u1 , u2 are two vector. W= span {u1, u2 } So, {u1 , u2 } is a basis . Now we found a basis ,…

Q: Show that S = {(i, 0, 0), (i, i, 0), (0,0, i)} is a basis for C

A:

Q: 2 then the is the matrix of transition from the base B to the basis B' and the coordinates of vector…

A: Given that B and B' are bases of a vector space v and P is the matrix of transition from base B to…

Q: Define a basis v1, v2 of R² by vị obtained by applying the Gram-Schmidt Procedure to v1, v2? 2 by…

A: Consider the provided question, A Basis of a vector space. Let V be a subspace of Rn for some n. A…

Q: 9. Find a basis for the hyperplane a where a-(2, 1, -3).

A: Any two linearly independent vectors which are orthogonal to the given a are forms a orthogonal…

Q: What is the maximum of || Ar||∞ if ||r||∞ = 1? What is the maximum of ||Ar||2 if |-|2 = 1? For…

A: A=2123 (a) We know that Ax∞≤max∑ka1k,∑ka1k,…∑ka1k when x∞=1. Ax∞≤max2+1,2+3=5 The maximum of Ax∞ is…

Q: Show that the vector space P, has the basis S = {1, x, x², x³}. %D

A: We have to solve given problem:

Q: 10. Consider the plane 2x - 3y+4z 0. Find a basis B of this plane such that (B = for ū =

A: Solution:-

Q: Find the orthogonal decomposition of v with respect to W

A:

Q: Compute the orthogonal projection of i = 1 onto the line L through 4 and the origin. projz (5) =

A: Vector projection

Q: Suppose {v1, v2, V3} is a basis for R³. Span {v2 – V1, V3 – V1} is a plane in R³. [Hint: What can…

A:

Q: Let {v1, ... , vn} be an orthogonal basis for Rn and let W = span(v1, ... , vk)· Is it necessarily…

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Q: When A is multiplied by a plane rotation Qij, which entries of A are changed? When QiJA is…

A:

Q: If P is the projection matrix onto the line through () -- (;) C) and ) then I – P projects onto the…

A: I have checked one by one option wise

Q: Let B'- {1+2x, 3-x} And B = {1,x} be Basis for Pi a) s Find the Transilion matrır Pine from B' toB…

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Q: Find a basis for the plane 4x + 3y + 5z = 0 in R³.

A:

Q: Explain why S is not a basis for P2. S = {1 – x, 1 – x², -1 – 6x + 7x²} O sis linearly dependent.

A: Given that S=1-x,1-x2,-1-6x+7x2 Explain why S is not a basis for P2 if S is linearly dependent the S…

Q: find basis and the 1) dimension of w ={A ¢ Mnz: A +=0}

A:

Q: D Consider the lineer tra given by Tlatbx+cx² lyou do not have to pro Find the folloceing a) ronge T…

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Q: Suppose T E L(V, W), and w1, ..., wm is a basis of range T.

A: Given that T∈L(V,W)andw1,.......wm is a basis of range T.To prove there exist…

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.