Let M2x3(C) be the vector space of 2 x 3 matrices over C, with the usual rules of addition and scalar multiplication, and also view C as a vector space over C with the usual rules. a12 a13 Let T : M2x3(C) → C be the map H a12- a21 a22 a23 Show that T is linear.

Q: Let V denote the set of all 1×3 matrices over R, and define usual addition on V as for M1×3. If the…

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Q: Let V be the vector space of symmetric 2 x 2 matrices and W be the subepace w - q W- span Find a…

A: First we calculate the basis of V.

Q: Determine whether the set of all matrices of the form (:: ;). a bc 1o), where a+e 0, and b +d+f 0,…

A: Given: abcdef , a+c=0, b+d+f=0

Q: Let M2x2 be the vector space of all 2 x 2 matrices, and define -[: :} T: M2x2 → M2x2 by T(A) = A +…

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Q: 4. Write the kernel of the transformation L(x) = (2x1 - x2, X2 + x3). Write the basis and dimension…

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Q: Let V be an n-dimensional vector space over F, and T:V→V be a linear transformation of rank 2. Prove…

A: T has rank 2 implies, for any two ordered bases the matrix representation of T has rank 2.

Q: T = e M2walc = 1 is a vector space under standard matrix addition and scalar multiplication. ulgn.…

A: The answer is given as follows :

Q: 2. Let P2(C) be a vector space of polynomials of degree less than or equal to 2 over R. (a) By using…

A: A linear transformation is a linear function from a vector space to another vector space. Kernel of…

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: 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: T= is a vector space under standard matrix addition and scalar multiplication

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Q: Let BV = {v1,v2} and BW = {w1,w2} be bases for vector spaces V and W, respectively. Let T : V → W be…

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Q: If T: V → W is a linear transformation of finite dimensional vector spaces and Rank(T) = 3, then the…

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

A: 'T' is a linear transformation if : 1. T( P+Q )= T(P) + T(Q) 2. T( aP ) = aT(P) ; where 'a' is a…

Q: Let V={a0+a1 x+a2 x2 | a0, a1, a2∈R} be a vector space of polynomial up to degree three, and…

A: We know that by definition of inner product x,y≥0 Take px=1+x+2x2, qx=1+x+3x2 then , p0=1, p1=1,…

Q: Let V be a 2-dimensional vector space with two bases B and B' and let T be a linear operator on V.…

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Q: Let be the vector space of polynomials of degree less than or equal to one with real coefficients…

A: The given vector space is the set of all polynomials of degree less than or equal to one with real…

Q: 5. Find the standard matrix corresponding to the linear map T : R2 - R? given by T(r, y) = (r+y,…

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Q: Let E= {e,, ez, ez} be the standard basis for R?, B = {b,, bz, b3} be a basis for a vector space V,…

A: a. Compute

Q: Let E=(e,,e2,e3) be the standard basis for R³ and B= (b1,b2,b3} be basis for vector spaces V, and…

A: Given the linear transformation T:R3→V s.t.  Tx1,x2,x3=x2-x3b1+x1+x3b2+x2-x1b3, where E=e1,e2,e3 is…

Q: · Let B = {b¡, b2, b3} be a basis for a vector space V. Find T (3b¡ – 4b2) when T is a linear…

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Q: Recall that a set of vectors 1. V2, in a vector space V is linearly dependent if there are scalars…

A: Using row reduced echelon form,

Q: Let p(x) = 3 - 2x and q(x) = 1 + x + x2              (p(x), q(x)) is the inner product on the vector…

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Q: Let M be the vector space of 2 x 2 real-valued matrices. For M = [ a b c d] we define M# = [d b…

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Q: Determine whether Tis a linear transformation (by proving or by giving counterexample) where P2 is a…

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Q: Let V be a vector space with basis B, let V ' be a vector space with basis B' , and let T :V → V' be…

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Q: T = 1 is a vector space under standard matrix addition and scalar %3D multiplication.

A: Given a set of matrices T = abcd ∈ M2x2 | c=1 To identify if this is a vector space or not.

Q: Let M be the vector space of symmetric 2x2 matrices, the vector space R² and the transformation S: M…

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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)…

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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: Let V be an n-dimensional vector space with ordered basis a, ß and y. Let C be the change of…

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Q: 4) Let : R³ → R[t]<2 be the linear map that sends 4(e₁)=t+t² 4(e₂) = 1-t 4(e3) = t Write the matrix…

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Q: 3). Determine whether the set of all matrices of the form a b c de f where a +c= 0, and b+d+ f = 0,…

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Q: 2. Let P2(C) be a vector space of polynomials of degree less than or equal to 2 over R. (a) By using…

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Q: a b Let M2 x2 be the vector space of all 2 ×2 matrices, and define T : M2 ×2→M2×2 by T(A) = A + A',…

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Q: Let P2(C) be a vector space of polynomials of degree less than or equal to 2 over R. (a) By using…

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Q: Let B = {b1, b2} be a basis for a vector space V. Let T: V → V be a linear transformation with the…

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Q: Let V be a 4-dimensional vector space, and let B={v1,v2,v3,v4} be a basis for V. Let T: V→V be the…

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Q: Let P2(x) be the vector space of polynomials with real coefficients of degree at most 2 with the or-…

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Q: Let V be the vector space of 2 x 2 matrices over R. Determine whether the matrices A, B,C E V are…

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Q: Let {v1, . . . , vn} be a basis for a vector space V, and let L1 and L2 be two linear…

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Q: Let B = {b, b, b3} be a basis for vector space V. Let T:V→V be a linear transformation with the…

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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: Let A = {a1, a2, a3} and B = {b¡, b2, b3} be bases for a vector space V, and suppose aj = 4b1 – b2,…

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Q: Let V be a vector space of dimension n, and let B and C be bases for V. Then the product PB-c Pc-B…

A: Step:-1 According to a well known result,  Let V be a vector space of dimension n and C and D be two…

Q: Let B = {b, b2, b3} be a basis for a vector space V. Find T(10b1 + 3b2) when T is a linear -6 1…

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Q: Let V be a vector space, and T : V →V a linear transformation such that every v e V is an…

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Q: Let V be a vector space on R of dimension 4, and T be a linear operator on V. In case that there…

A: i uave used the definition of diagonalisation

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: 4. Consider the linear map f : R1[X] → R1[X] given by f(ao + a1X) = (2ao +3a1)+ (ao – a1)X. Find the…

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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.