1. (a) Let U {A € M3 (R) | AT = -A}, a subspace of M3 (R). (i) Find a basis B for U. Justify your answer. (ii) Suppose that C = {C1, C2, C3} is another basis for U and that -5 -1 PB-C = 7 -8 -3 -4 3 1 Find C1. Justify your answer.

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: 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: (a) Consider the real linear map T : R' - R', expressed as follows: (1 2 -1 49 -1 X1 X1 + 2x2 - x3…

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Q: , find the change-of-coordinates matrix from the basis B= {1– 2t +t?,3 - 5t + 41,2- 2t + 5t?} to the…

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Q: et B = · 9, (x − 1)², 3x³) be an ordered basis for P3. Find the coordinate vector of (x) = 3x³ + 2x²…

A: Coordinate vector

Q: (a) Let A (b) Let A = = x = 1 1 arbitrary mx n matrix and x = = What are the row vectors and column…

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Q: Section 4.5 - The Dimension of a Vector Space Prereqs: Basis (4.3), coordinate vector and coordinate…

A: The given vectors are .The aim is to prove the given vectors are basis.

Q: Dases C= and E on R3, (i) Find the vector v3 of R' whose component vector with respect to C is [v3]c…

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Q: Let B = vector of 7 = E [V]B 0 1 00 {[%] [83] [28] [89]} 9 9 relative to B. -5 -2 || be an ordered…

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Q: Write standard Basis of P4(x)?

A: P4(x) is a collection of polynomial degree atmost 4

Q: y Suppose E (A, B) is controllable. Suppose a change of basis is performed so that we obtain a new…

A: As per the question we are given a controllable system as : ∑ ➞ (A, B) And after a change of…

Q: The set of solutions of the system of linear equations x₁ + x2-x3-4 = 0 x1 + x2 + ix3 + ix4 = 0 is a…

A: We have to solve given problem:

Q: 2) Let T a d 11 la + 2b + (-6) c + 2d + (-18) e 2a + 5b+ (-14) c + 6d+(-46) e Then a basis for…

A:

Q: 5. Let u= w= [-]-[], and w (a) Show that w is not a linear combination of u. (b) Show that u and V…

A: We have to show that-w is not a linear combination of u,u and v are linearly independent,w is a…

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…

A:

Q: The set B = {1+t², t + t², 1 + 2t + t²} is a basis for P₂. Find the coordinate vector of p(t) = 1 +…

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Q: Let p₁ (t) = 7+t², P₂ (t)=t-2t², P3(t) = 2+t-4t². Complete parts (a) and (b) below. a. Use…

A: The polynomials p1t, p2t and p3t belongs to the vector space P2 because P2 contains all the…

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

A:

Q: Solve the D. E. to the I. V. P. sinx(e-y + 1)x' – cosx = 1, y(0) = 0 %3D %3D (1 – e') (1 + cos.x) =…

A: We will find out the required solution.

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: Find a basis for the solution space  x1+3x2+4x3=0 3x1+8x2+7x3=0

A:

Q: 5- Suppose that A = {x1 = (3,1,2), x2 = (0,0,2), x3 = (2,0,1)} then [A] = R³, that means for every…

A:

Q: |a,6,d e а a, b, d E R}. S2x2 b d Do three matrices 1 A1 = 0 1 1 0 A2 = A3 1 0 - 1 | form a basis of…

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Q: Nap Lajoie has a lifetime batting average of 0.338. Assume that Nap Lajoie came to bat officially…

A: Given : Nap Lajoie has a lifetime batting average of 0.338 and came to bat officially six times…

Q: 5. Explain why three linearly independent vectors u,v,w in R³ form a basis for R³. (Hint: Consider…

A: Basis of a vector space :                               A non empty sub set S be a basis of 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 2. (а) Do the vectors vị = -1 -1 form a basis for R³? , V3 = 1 > V2 = -I--- (b) Do the vectors vị…

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Q: 1 Defined in P,(R), p(x)=1,q(x)=x- and h(x) =x² -: -x+- 6. Investigate whether the vectors are…

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Q: Problem 10. Consider the following set of polynomials in P₁: S = {1+ 3t² +t³, 1+t+ 2t³ + 3t¹, t+t² +…

A: We know that two vectors are linearly dependent if one vector can be written as a scalar multiple of…

Q: r projection oE the following

A: we've to show whether the projection operators are positive operators or not. Now a projection…

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…

Q: (a) Prove that for every positive integer n, one can find n + 1 linearly independent vectors in…

A: (a) To prove that for every positive integer , one can find linearly independent vectors in…

Q: V has bases B = (v1, v2, v3) and B' = (w1,w2, W3). If wi vị + 2v2 + 4v3, wz = 2v1 + 5v2 + 6v3, wz =…

A: Given that, B=(v1,v2,v3),B'=(w1,w2,w3) Given that,…

Q: [ -7 4 -4 4 -1 3 -7-4-1-6] A nonzero vector with integer entries in the null space of T'is Let T =…

A: Given: T=-74-4413-7-4-1-6 To Find: Non-zero vector of null space of T.