5. Consider the basis aj = (1, 2, 3), az = (2, 4, 5), and az = (3, 5, 6) for R] × 3. Consider t E L(R] × 3, R] × 3). If 3 Mtx, „(r) = find Mtx, (7).

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: 2, Xz} is a basis of V, (R). show that {, +X, X2 + X3, X3 +Xi} is also a basis of V, (R),

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Q: Apply the Gram-Schmidt process to find an orthogonal basis for S. 3 S = span 4

A: In this question, we find the orthogonal set of basis by the Gram-Schmidt process.

Q: How are n, rank A, and nullity A related to each other how can one find a basis for Col A, Nul A,…

A: Write the relationship between n, rank A and nullity A. Also write a procedure to find a basis for…

Q: In Rª, let S = {(1,2,2, –1), (0, –1,4, 2)} and T = U = sp(S) and W = sp(T). Find a basis for U +W…

A: Your approach is absolutely right!

Q: Explain why S is not a basis for P2.S = {−3 + 6x, 3x2, 1 − 2x − x2}

A:

Q: 1 Which of the following is a basis of R4that includes the vectors and 1 ? 1 -1 1 1 1 1 1 1 -1 1 1 1…

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Q: Explain why S is not a basis for P2.S = {2, x, 3 + x, 3x2}

A: From the given value for S to find the values for pi(x).

Q: 3) If V = R? and neither a, c nor b, c is a basis of R2, show that c 0. %3D

A: 3. It is given that a, c  is not basis of V ⇒c=λa    it is also given b,c is not basis of V⇒c=μb…

Q: et a1, a2 and a3 be colums of A, i.e A = [a1,a2,a3] and let W = span {a1,a2}   Find an orthonormal…

A: Given A=134203166 To find an orthonormal basis of W and distance between a3 and W.

Q: It can be shown that the vectors u1 = (2,−3, 1), u2 = (1, 4,−2), u3 = (-8, 12,−4), u4 = (1, 37,−17),…

A: A basis is a subset of a vector space that spans the vector space and is linearly independent.

Q: 個- to Ez in R². 3 a) Compute the change of basis matrix from 8 %3D

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Q: 1. Show that the following is a basis for R4:

A:

Q: 4. Present the standard basis vectors e, and ez in R? as linear combinations of (3, 1) and (1, 3).

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Q: . If {x1, X2, X3} is a basis of V3 (R), show that {x, +x2, X2 + x3, X3 + X1} _is also a basis of V3…

A: Given problem is :

Q: Explain why S is not a basis for P2.S = {1 − x, 1 − x2, −1 − 2x + 3x2}

A:

Q: 1

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Q: Find a basis fo

A: Given,            A=3-916-26-161-38

Q: Show that the set S= {(-2,-4), (1, 2)} is the basis of R2 or not?

A:

Q: 3) Let S contain vectors of the form f(x) = a + bx + cx² + dx³ : f(-2) = 0 and f'(-4)= 0: Then a…

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Q: 3. Let A {v1, v2, ..., Vn} be a basis for pV and n 2 2. Is B = {v1 + v2, , v2 + V3, ..., Vn + Vı}…

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Q: Take L: V --> W, from P2 --> P2 and L(p) = p + p', with basis B = {1 + x + x2, 1 + x, 2x + x2} and…

A:

Q: Show that {e1,€2, - } is a Schauder basis for P, 1<p< ∞, where S1 ifi=n (E") en = %3D 0ifi#n.

A:

Q: 3) li) Find the dual basis {l, l} of the standard basis of IR". (i) Fiod th

A: Since you have asked more than one question, we will solve first one for you. If you want other…

Q: Extend {l + x, 1 + x + x2} to a basis for P 2.

A:

Q: 5. Use the fact that {eina} is an orthonormal basis for L2(-7, T) to deduce that {cos(nx)} {sin(nx)}…

A: Consider the equation

Q: Suppose R* = is a basis for R*. Span {v1,..., V4}. Explain why {V1,...,V4} %3D

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

Q: Let A = {v1, vz, - also a basis for pV? How about the converse, i.e. if B is a basis for FV, is A…

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Q: Explain why S is not a basis for R2.S = {(1, 2), (1, 0), (0, 1)}

A:

Q: Explain why S is not a basis for P2.S = {−x, 4x2}

A: Given: S={−x, 4x2} To Explain: Why S is not a basis of P2.

Q: Show that S = {(i, 0, 0), (i, i, 0), (0,0, i)} is a basis for C

A:

Q: Consider the ordered basis 3 = {1, 1+ x, 1+ x + x²}. Given that %3D [1+ 2л + За?]8 — (а, b, с)*,…

A: Given β=1,1+x,1+x+x2 And 1+2x+3x2β=a,b,ct We have to find value for a,b and c…

Q: If {f1, . . . , fn} is an orthogonal basis of R^n and U = span{f1, . . . , fm}, show that U^⊥ =…

A: We will be proving the statement using basic definitions of Orthogonal Basis of Rn .

Q: 4.13 Let {u1,u2,..., un} be any orthonormal basis of (where F R or F = C). Show that %3D Euju; = 1.…

A:

Q: Let V = span {1+ 2x – x² + 3r³, -2+x+x² – x*, -1+ 8r – x² + 7r", 3 + x – 2r² + r'}. Find a basis ß…

A: Step:-1 Note:- To span V we need the polynomials of degree 3, so dimension of V is 4. (think using…

Q: Show that the set S = {p1.P2-P3} is a basis for P2 (space of polynomials of degree less or equal to…

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Q: 3. Compute dim(P3) using a basis that's not B = {r³, x², x, 1}. Hint : Let S= {2x3, r², x, 1}. Use B…

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Q: 4 2. If a basis for R3 is B = -3 and given [x]B find i in the standard basis.

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Q: Let B'- {1+2x, 3-x} And B = {1,x} be Basis for Pi a) s Find the Transilion matrır Pine from B' toB…

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Q: 4 2 If a basis for R3 is B = and if a vector in the standard basis is i = |-4|, find its 5…

A: We know, The set B of vectors in a vector space V is called a basis if every element of V can be…

Q: 23. Suppose R* = Span {v1, ..., V4}. Explain why {v1,..., V4} is a basis for R4.

A:

Q: 2. Construct a basis for T2,2(R²) and prove that it is a basis.

A: To find the basis of I2×2ℝ2. The set of all 2×2 matrices over ℝ. Let us claim that…

Q: Determine whether it is a basis. S= {4 – x, x³, 6x², x³ + 3x, 4x – 1}

A: No it is not a basis  see below the explanation

Q: find basis and the 1) dimension of w ={A ¢ Mnz: A +=0}

A:

Q: Suppose T E L(V, W), and w1, ..., wm is a basis of range T.

A: Given that T∈L(V,W)andw1,.......wm is a basis of range T.To prove there exist…

Q: 1- The vectors x, = (2,1, -4), x2 = (-3,4,1) and x3 = (2, –1,3) in R3, are %3D A: represent a basis…

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Q: S = (1+ x, 1- 2x) is not a basis for P,(th

A:

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.