Let T be a linear operator on a finite-dimensional vector space V. Prove the following results. (a) N(T∗T) = N(T). Deduce that rank(T∗T) = rank(T). (b) rank(T) = rank(T∗). Deduce from (a) that rank(TT∗) = rank(T). (c) For any n ×n matrix A, rank(A∗A) = rank(AA∗) = rank(A).

Q: Let A - ( ) a. Show that the vectors lz.A, A² are linearly dependent in Maz (R). b. Deduce that A"…

A: Statement (S1) proves that  I2 , A, A2  are linearly dependent.

Q: Let A be an m × n matrix. Show that every vector v in Rn can be written uniquely as w + u, where w…

A: Given: A is a m×n matrix To show that,  every vector v in Rn be written uniquely in the form, v = w…

Q: Find a basis for the kernel of the linear operator T(x, y) = (4x - y, -16x + 4y), Using the…

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Q: Find the standard matrix for the linear operator T:R’ → R° that first rotate a vector…

A: Let v1 = (x, y, z) be an arbitrary ℝ3 vector. If we rotate the vector counterclockwise about the y -…

Q: In the vector space P, of polynomials of degree atmost two, the first column of the matrix of…

A: The first column of the transition matrix obtained from the expression of first vector a1 with…

Q: Let, T: R -R; 7(x, y, z) = (x - y, 2x+3y + 2z, x + y +22) Find all the eigenvectors of the linear…

A: Given that, T:R→R; 7x, y, z=x-y, 2x+3y+2z, x+y+22 But consider the above is T:R→R; Tx, y, z=x-y,…

Q: In the vector space p of polynomials of degree atmost two, the first column of the matrix of…

A: We have to choose the correct answer from the given options according to the given information

Q: Let x and a be vectors in R2 Define the L: R2→ R2 by L(x) = x +a Is this function a linear operator?…

A: Linear operator should satisfy L:Rn→Rn  a) Lu+v=Lu+Lv         u,v∈Rn b)Lcu)=cL(u

Q: Show M={A in M2,2| |A| =0} is not a vector space under the usual addition and scalar multiplication…

A: We will show M={A in M2,2| |A| =0} Is not vector space by exhibiting two elements A,B belongs M but…

Q: Find a basis B for the domain of T such that the matrix for T relative to B is diagonal. T: R2 →…

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Q: Suppose that you have three vectors: f1 (x) = 1, f2 (x) = x – 1, and f3 (x) = } (x² – 4x + 2), that…

A: Given:fifj = ∫0∞fixfjxe-xdxf1x = 1f2x = x-1f3x = 12x2-4x+2D = ddx

Q: Q2) Let M2x2 be the vector space of all 2X2 matrices, and defined T:M2x2 → M2x2 , such that T(A)=…

A: T:M2×2→M2×2, such thatTA=A+AT

Q: a) T: M, → R, where M, is the vector space of n by n matrices over a field, and T (A) = det(A).

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Q: Let T be a linear operator on a two-dimensional vector space V and suppose that T≠c| for any scalar…

A: Let T be a linear operator on a two-dimensional vector space V.

Q: Find a basis B for the domain of T such that the matrix for T relative to B is diagonal.T: R2→R2:…

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Q: (d) Let V = R³, W = R² and define T : V → W by T: (x1, x2, x3)(x₁ - x2, X2 - x3). Find the kernel of…

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Q: 8. Let P3(C) be a vector space over C and let D: P3(C) → P3(C) be the differential operator defined…

A: D: P3ℂ→P3ℂ be the differential operatorDpt=dpdtrelative to the basis 1, t, t2, t3

Q: In the vector space f, of polynomials of degree at most two, the third column of the matrix of…

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Q: Find the coordinate vector ofp(x) = 1 + 2x + 3x2 with respect to the basis B = {l + x,- x, x2} of…

A: Given :           p(x) = 1 + 2x + 3x2            B = 1 + x, - x, x2 of P2

Q: 3. (a) Let S = {v1, . . . , vn} be a linearly dependent subset of a vector space V , and let vn+1 be…

A: This question is related to linear algebra.

Q: In the vector space P,. of polynomials of degree at most two, the first column of the matrix of…

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Q: In the vector space P of polynomials of degree at most two, the second column of the matrix of…

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Q: Find a basis B for the domain of T such that the matrix for T relative to B is diagonal. T: P1→P1:…

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Q: Let V be a finite-dimensional vector space over F, and suppose that T in L(V, V) has the property…

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Q: Conclude that there is a vital assumption when using rank nullity, often overlooked. What is it? Let…

A: According to Rank-nullity theorem, if V and W are vector spaces and T is a linear operator from V…

Q: Let Give a non-zero vector in the null space of A. t= A = 1 -4 3 1 -1 -3 –1 4 -2 -2 2 -5 -5 1

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Q: Find a basis B for the domain of T such that the matrix for T relative to B is diagonal. T: R² → R²:…

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Q: If M = {x: Lx = 0}, where L is a continuous linear functional on H, prove that M¹ is a vector space…

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Q: Define T: R-R by Tix) =Ax, where Ais the matrix defined below. Find a basis B for R? with the…

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Q: Find a basis for the vector space {A E R?x2 | tr(A) = 0} of 2 x 2 matrices with trace 0.

A: Basis of vector space V: A subset S of a vector space VFis said to be a basis of VF,if 1. S contains…

Q: a) T: M, → R, where M, is the vector space ofn by n matrices over a field, and T (A) = det(A). b)…

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Q: On the vector space E = Mn(R) of all square matrices, let N(A) = n max |a,l A = (as) € X. Prove that…

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Q: Let B= {b1, b2} and c= {cl, c2} be bases for a vector space V, and suppose b1 = - cl + 4c2 and b2 =…

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Q: In the vector space P of polynomials of degree atmost two, the third column of the matrix of…

A: The solution is given below in the next step:

Q: Let B and C be bases for a vector space V and let the change-of-coordinates 2 -3 matrix from B to C…

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Q: Let ₹₁ = [1; 0; 0], ẻ2 = [0; 1; 0], ẻ3 = [0; 0; 1] denote the standard vectors in R³, and let T: R³…

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Q: In the vector space P, of polynomials of degree at most two, the third column of the matrix of…

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Q: Let W be the vector space of 3x3 symmetric matrices, A E W. Then, which of the following is true? a)…

A: Given that W is a vector space of 3×3 symmetric matrices. i.e W=A∈M3×3| AT=A Let, A∈W where,…

Q: a) T: M, → R, where M, is the vector space of n by n matrices over a field, and T (A) = det(A). b)…

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Q: (a) Find the matrix representation of L(x), the linear operator on R? that first rotates by 30°…

A: Vectors.

Q: Q3. Let L: R2R2 be a mapping from the vector space R2 to itself defined by L(x , y)= ( 2x+y, x-3y)…

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Q: a) T: M, → R, where M,n is the vector space of n by n matrices over a field, and T (A) = det(A). b)…

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Q: Find the standard matrix A and A' for T = T2 • T1 and T' = T, ° T2, where T1:R2 → R³,…

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Q: In the vector space R2 consider the two bases given by [u1, u2] with u1 = (1, 2)T, u2 = (2, 5)T and…

A: To  find the transition matrix Given, u1=12 , u2=25 , v1=32 , v2=43  The transition matrix of u1,u2…

Q: Define a linear T: R₂[x] → R3 [x] by T(p(x)) = x²p"(x) − 2p'(x) + xp(x), where p'(x) and p''(x) are…

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Q: Label the following statements as true or false. (a) Suppose that B = {x1,x2, ..., xn} and B' = {x1,…

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Q: S. The image of the vector i + j – 2k under a mapping (1 2 3) defined by the matrix M, where M =0 1…

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Q: Let D= {d1,d2, d3} and F= {f1, f2, f3} be bases for vector space V, and suppose f1 = 2d1 – d2 + d3,…

A: Change-of-coordinates matrix from F to D is,

Q: a) T: M, → R, where M, is the vector space ofn by n matrices over a field, and T (A) = det(A). b)…

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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.