3 Let M₂(R) be the vector space of all 2 × 2 matrices with entries from R. Show that M₂ (R) = W₁ W₂, where W₁ a {(" "); a,bER} and W₂ = {(J)ieder} R

Q: 2. Let P2(R) be the vector space of polynomials over R up to degree 2. Consider B = {1+x= 2x°, -1+a…

A: given B=1+x-2x2 , -1+x+x2 , 1-x+x2B'=1-3x+x2, 1-3x-2x2 , 1-2x+3x2 b) find the coordination matrices…

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: Which of the following cannot be defined as an inner product between two vectors (x0, Yo) and (¤1,…

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

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Q: If vectors b, c and d are not coplanc then prove that vector

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Q: (1,1,..., 1)T be the column vector consisting of all 1s. Prove 0. 3. Let a = that La

A: Given: Let a=(1,1,......1)T  Then objective is to prove  La=0

Q: Suppose that f is a vector of length 8 with F1{feven} = -2 – i and F{f»dd} = 3+i. %3D Compute the…

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Q: (a) Let A, B be n × n symmetric matrices such that XTAX = X™BX for all column vectors X e R". Show…

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Q: 3. Let vi - [1,-2,3], va -(-1,-4,5], and va [5,-1,3] be vectors in R'. a) Use determinant to show…

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Q: The equatiokn  Y = (x,y)T  is given as a vectorfunction with 2 components and A = table below 3 2 2…

A: Given is a matrix A and we have to diagonalize the matrix A. First we find eigenvalue of matrix 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…

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Q: Let the matrices Ho, H, Ha 1x1 M어ni x [1]. be defined ... the He=L He-I vector of length n-2" can be…

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Q: Suppose M is the set of all real column vectors in R' whose components are equal, then the basis for…

A: See the detailed solution below.

Q: 14. Find a vector perpendicular to both of the vectors * (ũ + v) and (ũ – v) where ủ = 2i – j + k,…

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Q: (2) Decide whether the given statement is true of false. Prove your answer. i) If Az = Bx for some…

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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: Let u be a vector in R" and A be an m × n matrix. If u is orthogonal to each row of A then u is in…

A: Since u is orthogonal to each row of A, it means u is in null space of A.

Q: 20. Find two vectors of norm 1 that are orthogonal to the vectors u = (2,1,-4,0), v = (-1,–1,2,2),…

A: To find two vectors of norm 1 that are orthogonal to the below vectors.…

Q: 10. Show that V = R² with the standard scalar multiplication, but addition defined by (7, , v.) +…

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Q: Consider the vector space M consisting of all matrices of the form b } a c 3a +4b-3c where a, b, and…

A: Given that M is spanned by abc3a+4b-3c|a,b,c∈ℝ If -34-29/5122/5 is in M then it must be of the…

Q: 4 [1] 2. Is the vector 5 in the column space of matrix A = 1 0 -2 −4 4 4 5 3 -13]

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Q: If a, and b are non - collinear vector and vectors a = (x +4 y)a +(2x + y +1)b and, B= (-2x + y + 2…

A: We will solve the following

Q: r projection oE the following

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

Q: 6. Let B be the following ordered basis for R?: в %3 {(1,—1), (2, -3)} . Find the coordinate vector…

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Q: Let v be an n-vector and w be an m-vector. a) What is the matrix product (a';) = vw? Give a formula…

A: Given : v is an n-vector and w is an m-vector To give formula for aij in each of the following…