Let A E Mpxn(F), and let B be the set of rows of A. Prove that B is a basis of F" if and only if det(A) + 0. Conclude that det A = 0 if and only if one row is a linear combination of the others.

Q: e set B = {v= (1,1,1 a basis of R. 3. Linearly depende

A: Given set B = { v = ( 1, 1, 1 ), u = ( 0, 0, -3 ),  w = ( 2, 2, 0 ) } Then B is: -  A. a basis of R.…

Q: 2) Show that S ER° defined by S = {(1,0, 5), (0, 2, -1)} is linearly independent. Extend S to be a…

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Q: Let β = { u1, u2, .. . , un} be a subset of Fn containing n distinct vectors, and let B be the…

A: Given that, β=u1,u2,...,un is a subset of Fn and let…

Q: 4 Let A = and define T : R? → R? by T(x) = Ax. Find a -2 -1 basis B such that the matrix [T]s is…

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Q: Let Find an orthonormal basis of the row space of A. A = 1 4 1.[C ]]}.

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Q: Determine whether the set (a) spans R3, (b) is linearly independent, and (c) is a basis for R3.S =…

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Q: Explain why S i not a basis for P2. S = {1, 4x, -5 + x?, 3x} O sis linearly dependent. O s does not…

A: We have to find out

Q: Let B be the basis of R² consisting of the vectors and let C be the basis consisting of {B} Find a…

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Q: 1 Let u₁= = U₂ = U3 = 1. and y 3 3 1 (a) Show that {u₁, U2, U3} is an orthogonal basis for R³. (b)…

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Q: Let B be the basis of R² consisting of the vectors and let C be the basis consisting of Find a…

A: Solution:-

Q: Let B be the basis of R? consisting of the vectors 4 -2 and let C be the basis consisting of 3 2…

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Q: 2. Let B₁ be the basis 2²e², re², e² of V = span(2²e², zeª, z), and B₂ be the basis of that the map…

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Q: Let fi = 1+æ + 2x?, f2 = 2 + 3x + 3x?, f3 = 3 + 5x + 3x? and set F = (f1, f2, fs), a a13 a12 а11 a21…

A: We have to find transition matrix from S to F.

Q: 3. Determine whether the set S = {1 – t, t, 1 + t2} a. Spans P2 b. Is linearly independent с. Is a…

A: Given set is  S = { 1-t ,t , 1+t^2}

Q: Let S = {v1, v2} be a basis of IR? where vi = and What element of R? is determined by the linear V2…

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Q: In the vector space P, of polynomials of degree at most two, the matrix of transition from the basis…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Determine whether the set (a) spans R3, (b) is linearly independent, and (c) is a basis for R3.S =…

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Q: In part (a.) wouldn't you have to prove that U = Span(B) as well as B being linearly independent to…

A: Yes, we need to show both the parts. And the question you asked has already been proved in the…

Q: Let S be the standard basis for IR and consider the non-standard ordered bases of R4 1 3 -1 3 1 -2…

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Q: Let 3 the the standard ordered basis on M2×2(R), let Y = {1 – x, 1+ x}, and let T: M2x2 (IR) → P1…

A: a = 10 b = 2  The detailed solution is as follows below:

Q: the matrix of T relative to the basis B= {(1, 1), (- 1, 0)} and verify that [T], [a], = [T(a)]B Va e…

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Q: Let TilP > Pz, the basis of NIT) T (ax²+bx+c) = 2Qx+ b. Then %3D is

A: Applying nullity definition we have

Q: Find a basis for the subspace a W = E R2x2 | 6a + 2b – c = d of R2x2- A basis for W is 1 0 0 Ex: 1 :

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Q: Let T : R? → R´be defined by [2 _5||X1 X2 1 6] [x2] а If B = is a basis for R“ such that |T'|B is a…

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Q: Explain why S is not a basis for M,2" 0 1 1 0 S = 1 0 0 1 O sis linearly dependent. O s does not…

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Q: Consider the matrix -1 A = 1 1 Find a basis of the orthogonal complement of the null space of A. ||

A: We know that Null Space of A={ x∈ℝ2 : Ax=0 }where x=( x1 , x2 )Orthogonal complement of vector  x=(…

Q: Let B be the basis of R? consisting of the vectors and let C be the basis consisting of Find a…

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Q: In Exercise, find the orthogonal complement W1 of W and give a basis for W1.

A: Consider the line W = xy: 2x - y = 0 We can see that the here W is a line. W = xy: 2x - y = 0

Q: Find the dimension of the vector space V and give a basis for V.

A: v={A in M22 : A is upper triangular}

Q: Let β ={x1,...,xn} be a subset of Fn containing n distinct vectors, and let B denote the element of…

A: Assume that β ={x1,...,xn}  is a basis for Fn.Let B denote the element of Mnxn (F) whose jth column…

Q: Let B = {1+t,1 – t} and C = {4, 1 + 5t, t²}. Does B form a basis for P2?

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Q: Let B = {v1, v2, V3} be a basis for R3. For any u, v E R³ define || (u, v) = [u]E[v]B• (a) Is (, )…

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Q: Find a basis for the vector space (A E Rx4 tr(A) = 0} of 2x 2 matrices with trace 0. B = { }.

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Q: let "S={v1,v2,v3}" be a basis of vector space v, determine whether T={2 v 1,v1+v2,v1+v3} is a basis…

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Q: A is a 3 x 5 matrix and (v1, v2) is a basis of null(A). The vectors v3, v4, vz E R³ satisfy Avz =…

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Q: Given a basis B = {V1, V2,, Vn} of Subspace S, there is a unique way to express any ... vE S as a s…

A: We will first write 2 expressions of v as a linear combination of basis vectors in B. Then, we will…

Q: Find a basis B for the domain of T such that the matrix for T relative to B is diagonal. T: P2→P2:…

A: Given, the map               We have to find the basis B for the domain of T s.t. T relative to B…

Q: Let {v1, ... , vn} be an orthogonal basis for Rn and let W = span(v1, ... , vk)· Is it necessarily…

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Q: Explain why S is not a basis for P,. S = {1 – x, 1 – x², -1 – 6x + 7x2} || O sis linearly dependent.…

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Q: If {v1,..., Vn} is an orthonormal basis for R", then vịvi + · · ·+ vnvn = I. O True O False

A: Given, v1, v2, ..., vn is an orthogonal basis for Rn. then v1v1T+...+vnvnT=I It is True or False. To…

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

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Q: Explain why S is not a basis for P. S = {1, 2x, =7 + x², 9x} O S is linearly dependent.

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Q: 2. Consider the set (a) Verify that this is an orthogonal basis of R³. (b) Write the vector v = as a…

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

A: Given the basis B = e1=1+x ; e2=x+x2 ; e3=2+x2 the basis A = a1=1 ; a2=x ; a3=x2

Q: Let B the the standard ordered basis on M2x2 (R), let y = {1, x}, and let T : M2x2 (R) → P1 (R) be…

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Q: Explain why S is not a basis for P,. S = {1 – x, 1 – x2, -1 – 7x + 8x²} O sis linearly dependent. O…

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

A: Given : vector space P2, of polynomials of degree at most two To find : the second column of the…

Q: Let B = {v1, .. be a linearly independent set in Rn. Explain why B must be a basis for TR"

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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.