3. The purpose of this exercise is to introduce a method based on the Gauss elimination method in order to extract a basis out of a gener- ating set. Begin with the vectors: 1 HOO 5 -3 Let V be the subspace of R³, spanned by these vectors. We wish to find a basis of V. All we need to do is apply the Gauss elimination method to the matrix with columns the above vectors. The columns of the resulting matrix that have a pivotal elements will correspond to the columns of the initial matrix that are linearly independent and span the same subspace as all the columns together. Apply this to the above columns and find a basis for V.

Q: 2. How many vectors does any basis of of Max(a+2)(R) contain, if a = max1<i<10 a;?

A: Since, dimension of a vector space is equal to number of elements in basis of vector space.

Q: Write down a basis of R 4 where you don't use any of the standard unit vectors (e1, ... , e4). Show…

A:

Q: B = (1, x+1, x² + x) forms an ordered basis for P3. Give the B-coordinate vector of each of the…

A: The coordinate vector of v with respect to  is given by such that .(a) Given .Comparing the…

Q: 5. W is spanned by the vectors 24 -3 9 and Find a basis for W¹.

A: The basis for W is:Basis of W= 24-3-8 ,  -1-225

Q: Find a basis for the solution space of the given homogeneous linear system. X₁ 2x2 + 10x3 = 0 2x₁…

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Q: 14a.Show that the vectors w₁ = (0,2,0), W₂ = (3,0,3), W3 = (-4,0,4) form an orthogonal basis for R³…

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Q: 2.) Determine whether S = {G.1).(1.0).(2,12,6)} is a basis for R. If it is write u = (6,3,6) as a…

A: Determine whether S=23, 52, 1, 1, 32, 0, 2, 12, 6 is a basis for ℝ3. A set of linearly independent…

Q: A square n x n matrix is upper triangular iff all the entries below the diagonal are zero. In other…

A:

Q: 5.- Demonstrate whether the vectors of the following vector space are linearly dependent or linearly…

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Q: Express the vector (4, 7, -2) as a combination of the standard basis vectors i = (1, 0, 0), 3 = (0,…

A:

Q: Q3) Consider the following matrix A, 2 0 -2 -1 0 1 -3 3 |-2 1 A =| 1 1 |-2 0 2 0 -3 2 -1 3 i) Find a…

A: As per guidelines I have to solve only first three subparts. For next question please resubmit the…

Q: Make the orthagonal set of functions {1, cos(x), cos(2x), ...} orthonormal on the interval [0,2pi].

A: Two distinct functions f and g are said to be orthogonal on [a,b] if &lt;f,g&gt;=0   , where &lt; ,…

Q: (1 point) The standard basis & (e₁,e₂) and two custom bases B (b₁,b₂} and C (c₁, c₂} for R² are…

A: Solution:

Q: Let A =[ 4 -1 2 , -1 8 3 ,1 -2 5 ] b=[1 ,-2,3] and initial vector x 0=[0,0,0,0] perform  two…

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Q: 1a. Given vectors v₁ = (1, 0, 0) and v₂ : (1, 0, 0) and v₂ = (0, 1, 0) in R³, find a vector V3 such…

A:

Q: [5+7i] 3+4i b) A (2+9i) [5+7i 3+4i

A:

Q: Let B = (1, t + 8, (t – 1)², 4t° ) be an ordered basis for P3. Find the coordinate vector of P(t) =…

A:

Q: In P,, find the change-of-coordinates matrix from the basis B= (1-2t+P,3-5t+41,1 B-coordinate vector…

A: Given B=1-2t+t2, 3-5t+4t2, 1+4t2. To find P2 that transforms B to C, where C=1,t,t2 PC←B=131-2-50144…

Q: 1. Determine S = { -1}is basis for R°. If it is write the vector z = as a+ a linear combinations of…

A: We have S=abc,b+1c+1a+1,c-1a-1b-1 To determine the set is a basis of R3 or not.   The basis for a…

Q: 1. Show that the vectors (1, 1,0), (1, 1, 1) and (1,0, 0) are a basis of R3. Find the coefficients…

A:

Q: 1 3 1 2 A=2 6 4 002 4 I= I Y Z ພ () and 6 = 3

A: Solution to the system

Q: 3 0 1 -1] -3 X= -4 -4 1 x(0) 1 1 -2 5 -1 2 16 3.

A:

Q: 1-5 52 0-4 0 100 -7 0 0 10 -8 0 0 0 0 0 +X5 (Type an integer or fraction for each matrix element.)

A:

Q: 6. Use the vectors below to complete the following A-B V1 V2 = (a) Find an orthonormal bases for…

A:

Q: Find a basis for the solution space of the given homogeneous linear system. Xy - 3x2 + 3xy - 3x4 = 0…

A: The homogeneous linear system is x1-3x2+3x3-3x4=02x1-5x2+12x3-4x4=0The corresponding matrix form…

Q: c) Consider the vector X₁. -3 = [³3] i. Express x₁ as a linear combination of the standard basis…

A:

Q: 5. Determine whether the following three vectors v, = (1, 2, 3, 4), = (0, 1, 0, -1) and v, = (1, 3,…

A: To determine the given vectors form of basis for or not.

Q: 5. Determine whether vectors u =(2,-6,-5), v = (3,8, 18) and w = (8,-6,7) will form a basis for R³.…

A: Basis is very important in linear algebra. A basis set always contains linearly independent vectors…

Q: VII. For R3, find the Change of Basis matrix from basis B to basis D. -2 As a mild form of…

A:

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: If B= {(1,1,0), (0,1,1), (−1,0,2)} is a basis of R³, then the coordinate vector of v= (8, -5, 3) in…

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Q: 3 -3 3 A = 6 -9 a. A basis for the row space of A is { 3. You should be able to explain and justify…

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Q: 1. (a) Determine whether the following three vectors v, = (0, 3, 1,-1), v2 = (6, 0, 5, 1) and vz =…

A: Solution Given v1=(0,3,1,-1) v2=(6,0,5,1) v3=(4,-7,1,3) in R4 We need to find the linearly dependent…