C: Let T be the linear operator associated in the standard basis B of F2 with the matrix 2 4 -1 Let B' be the basis ((2, -1), (2, 1)) M = b) Using eachof these three matrices find the dimension of the range of T (you must check that the mension is the same with all the matrices)

Q: What is the scalar projection of vector a given by (a₁,a2,a3) in the direction of vector b given by…

A: We have to find the scalar projection of a in the direction of b.

Q: Explain why the vectors U₁=(3, 8, -8) U₂ = (-12, -32, 32) form a linearly dependent set of vectors…

A: Linearly dependent vectors

Q: Find a subset ofbthebvectors v1=(0,2,2,4), v2=(1,0,−1,−3), v3=(2,3,1,1) and v4=(−2,1,3,2)that forms…

A:

Q: Find the coordinate vector [x]gof x relative to the given basis B= {b1,., bn}. -[-]»= [-{]==(:)

A:

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

A:

Q: Determine whether the vectors (3,0, 3, 6), (6,0, 0, 3), (0,0,3,0),(6,3,9,12) form a basis of R*.

A:

Q: Show whether or not the set of vectors B = {U, V, W} where U = V = and W = 0 is a basis for R³.

A:

Q: "Write x as the sum of two vectors, one in Span (u,.u₂.3), and one in Span (U4). Assume that (u) is…

A: In this question , we will write the 'x' as linear combination of remaining vectors .

Q: Show linear dependence in part (ii) to show that ({1,0,1}) is a basis.

A: From part ii)We have to show that the set  is a basis of .We know thatA basis B of a vector space V…

Q: The vectors v1, v2 and v3 form an orthonormal basis for R3 where v1:=vector([0,0,-1]);…

A:

Q: 6. Let V be the vector space of all diagonal 5 x 5 matrices, then dim(V)=--

A: Given that the vector space contains all diagonal 5×5 matrices and we need to find its dimension.

Q: Find the basis for the orthogonal complement V: v1=(1,-3,3,5) v2=(2,-5,9,3)

A: The given vectors  and .The objective is to find the orthogonal complement of the given vectors.

Q: Show that the vectors (1, 1, 2), (1,2,5), (5,3,4) of R'(R do not form a basis of R'(R).

A:

Q: Let B = (1, x, x², x³) be an ordered basis for P3. Find the coordinate vector 8x³ + 2x²7x - 9…

A:

Q: -H 3 { and C { Let B } be two bases of R2 Find the change of coordinate matrix P_ sending a B…

A: Required to determine the change of basis matrix from B to C.

Q: Determine whether the set of vectors {(2,-1,3),(4,1,2),(8,-1,8)} form a basis for R3.

A: Consider the vectors : 2,-1,3, 4,1,2, 8,-1,8 To form a basis these vectors, the vectors must be…

Q: Decide whether the vectors [2, 9, -1, 4, 7, 8, 10, 9], [11, 2, 5, -9, 0, 0, -1, 3], [4, 18, -2, 8,…

A: We have to decide whether the given vectors are linearly independent. Then, we can proceed further.

Q: Find the coordinate vector of a = [*]B= -10-30- 2 with respect to the basis B = for R³.

A: Given: It is given that, x→=42-4 and B=152, 01-6, 001. To find: We have to find the coordinate…

Q: V2, V1 + v2, 2v1 - v2 ), v = (9, 9), w = (-3/2, 2, -8) (a) the image of v (b) the preimage of w (If…

A: This is a problem of vector calculus.

Q: Draw the set of all two-dimensional vectors whose 1-norm is equal to 2. For the matrix 7 A = -2…

A: Given: A=700-2

Q: Consider linear mapping A E L(Vn) and two basis X, Y of Vn. Select all true statements. Let Y be a…

A: Given that, Vn→vector space X=x1, x2, … , xn; Y=y1, y2, … , yn bases of Vn. A∈LV Verify the given…

Q: 2 COLAS -1 Consider the two bases B= for R² and C= P √2 a.) Compute the change-of-coordinates matrix…

A:

Q: Use matrix multiplication to find the orthogonal projection of (-8, 5, 7) on the xy-plane. Let T…

A:

Q: Represent the vector ⟨16,8,−14⟩ in terms of the ordered basis B={⟨5,−2,−2⟩,⟨1,−4,2⟩,⟨−1,−5,3⟩}.

A:

Q: Show that the vectors (4-3) and (5,2) form a basis CF TR

A:

Q: Determine the dimension of the vector space. M2,3 STEP 1: Determine the number of linearly…

A:

Q: 4. Let V = R³ and let T: V →V be the linear map given by T(r, y, 2) = (2x + z, y – x+ 2, 32) (you do…

A:

Q: a Consider the vector space W { : a + d = 0} of 2 × 2 matrices having trace c d 3 zero. Also let v =…

A: Please find the answer in next step

Q: Let V₁, V2, V3 be the vectors in R³ defined by 18 ---A V2 = -14 (a) Is (V1, V₂, V3} linearly…

A: We now solve this questions