2. Show that {1,x – 1, x (x + 1)} is a basis of the space of polynomials P2 (IR) and that W = {p(x) E P2 (R) : p (0) = 0} is a subspace of P2 (IR). Find the dim W.

Q: (2) Find a basis for the subspace W = {(a+c,2a – 6, 3a – 26 – c, b + 2c) | a, b, c E R} of R4

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Q: Show that W = {(x1, x2): x1 ≥ 0 and x2 ≥ 0}, with the standard operations, is not a subspace of R2.

A:

Q: 2. Consider the linear transformation T: P2(R) → P2(R), defined by T(f(x)) : f'(x) + 5f(x). Find the…

A: The linear transformation,   T:P2ℝ→P2ℝTfx=f'x+5fx

Q: For the linear map T = LA : R³ → R³ given by multiplication by A = 300 024 002. find the T-cyclic…

A: Let T be a linear operator on a vector space V and let x be a non zero vector in V. Then the…

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

A:

Q: For the linear map G : R³ → R² defined by G(x, y, z) = (x + y + z, 2x + 2y + 2z), find a basis and…

A: We have to find basis and dimensions of image and kernel of G.

Q: Let L = {(x1, x2, x3, X4, X5) E R° | x1 X2 + 2x4 = 0, x2 – 3x4 = 0}. | Prove that L is a subspace of…

A:

Q: 1. Suppose that A is an nxn non-zero, real matrix and 2 is a fixed real number. Let E ={x=R* : AT =…

A:

Q: 2. Let P3(R) be the vector space of all polynomials of degree 3 or less (and the zero polynomial).…

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Q: Which of the following is a basis for the subspace {:) X y z x = y+z and x, y,z €RC M2x3(R) ? 0 x 0…

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Q: Find a basis for the following subspace of IR°. P = {¤ E R° | x1= x3 = x5, x2 + x4 = 0} What is the…

A: Explanation of the answer is as follows

Q: let V be the subspace spanned by {w1,w2,w3} .Find a basis {v1,v2} of V such that neighter v1 nor v2…

A: Given: V is spanned by {w1, w2, w3}  w1 = [-1 4 5]t, w2 = [0 -4 0]t, w3 = [2 -20 -10]t Consider the…

Q: If V is a linear subspace of R", and x1 , ×2 , ... , Xn E V, then we must have span {x1, X2, Xn} =V.…

A: To determine that the statement, if V is a linear subspace of ℝn and x1, x2,....,xn∈V, then spanx1,…

Q: 7 4. 3 and let W the subspace of IR spanned by u and v. Find a basis of W, the orthogonal complement…

A: The objective is to find orthogonal complement of W

Q: 1. Determine whether w - {[:]•r:a-4-0} X1 E R? : : x1 – 23 = X2 is a subspace of R?.

A:

Q: Say whether the polynomials P1 (x) = x³–7x²+4x, p2(x) = 3x³-x²+5x-1, p3(x) = x³+4x-x²+1 are linearly…

A:

Q: Prove that W1 = {(a1, a2, . . . , an) ∈ Fn: a1 + a2 + · · · + an= 0} is a subspace of Fn , but W2 =…

A: Prove that W1 = {(a1, a2, . . . , an) ∈ Fn: a1 + a2 + · · · + an= 0} is a subspace of Fn , but W2 =…

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…

A:

Q: 3 Find a basis of the subspace of R' defined by the equation -7x1 – 5x2 – 4x3 = 0.

A:

Q: 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…

A: The proof of given question is solved below.

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 :

A:

Q: 4) Prove that the following subsets of R4 are subspaces, and for each subspace find a basis and the…

A:

Q: Consider P2(R) with its basis B = (x, x + 1, x² –- 1). Find [x2 + 3x + 5]B.

A:

Q: 3.12. Show that for a subspace E we have (E-)- = E. Hint: It is easy to see that E is orthogonal to…

A:

Q: If Sz is a subspace of R* of dimension 3, then there cannot exist a subspace S2 of R* such that S, c…

A: S1, S2 are subspaces of R4 such that S1⊂S2⊂R4 with S1≠S2 and S2≠R4.

Q: Find a basis for the subspace 96 4a + 6b W = E R2x2 a, b eR} of R2x2- 86 5a Ex: 1 4 6. A basis for W…

A:

Q: 2) Let W1, W2 be subspaces of V. Prove that W1 UW2 is a subspace if and only if W1 C W2 or W2 C W1.

A: Let W1,W2 be subspaces of V. To prove W1∪W2 is a subspace if and only if W1⊆W2 or W2⊆W1

Q: 3 Find a basis of the subspace of R’ defined by the equation -5x1 – 7x2 – 2x3 = 0.

A: Consider the given equation:

Q: The set of polynomials of degree exactly two is a subspace of P2.

A: False

Q: 4. Let H be the set of 2 × 2 whose trace is equal to zero. That is [a11 a12 Н- E M2(R) | a11 + a22 =…

A:

Q: Find a basis and the dimension of the subspace W of P3= {ax3+ bx2 + cx + d} defined by W = {PEP3:…

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Q: Prove that the the solution to the equation 2x – y+z= 0 is a subspace of R³.

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Q: Find the basis and dimension of the kernel and the image of G, where G(x, y, z, s,t) = (x + 2y + 2z…

A: First, we find the matrix corresponding to this transformation. Then apply row reduction to that…

Q: Let P, be the set of all polynomial functions in P(R) of the form ·+ anx", for some fixed positive…

A:

Q: 6. Find a polynomial p(x) for which {r² – 1, x² +x +2, p(x)} is a basis of P2(R).

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Q: 1. (- Let W₁ --E--E) and W₂ = = 1 -1 (a) Find a basis for W₁ W₂. What is dim(W₁ W₂)? (b) Show that…

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Q: () If S is a subspace in R that is spanned by then S is spanned by: O a. 1 and 0. 1 1 and 1 c. and…

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Q: Prove that W1={ (al, a2, ..., an)e Fn: al+a2+ +an=0} is a subspace of Fn, but W2= { (al, a2, ...,…

A:

Q: Let W1 and W2 be two subspaces of a finite dimensional vector space V over a field F. The formula…

A:

Q: 27. Find the dimension and one basis of the nullspace V = {(x1; x2; X3; T4) E R4 : x1 + x2 – x3 = 0…

A: As per our guideline we are supposed to answer only one question if multiple questions are posted.…

Q: 5. Consider the vector space P2(R), the set of all polynomial func- tions of degree less than or…

A:

Q: 7. Consider the subset S of R³, S = {(x, y, z) | 2x – y + 3z = 0}. Show that S is a subspace and…

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Q: 1- Show whether the set of polynomials a, +at +at+at ,where a, =a, +a, is a subspace of P3 or not ?

A: Let V be a vector space over the field F. A non-empty subset S of the vector space V is said to be a…

Q: 6. a. Give a basis for the orthogonal complement of the subspace V C R' given by V = {x € R* : xị…

A:

Q: (c) Let M = (xn)el² :x, + x, = 0}. Show that M is a closed subspace and find an orthonormal basis…

A: Please check step 2 for the solution.!

Q: Given that x, = – 1 and x1= 1. Then the basis of the first Lagrange polynomial Lo(x) and L,(x) are…

A:

Q: Which of the following is a basis for the subspace {(: X y z W = x = y + z and x,y,z e RC M2x3(R) ?…

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Q: Find a basis for the subspace {p(=) e P: p(2) = -p(-2) } of P. W = A basis for W is {x² -Ex: 1 :,…

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Q: 60. In R4, given a nullspace V = {(x1; X2; X3; X4) E R4|¤1 – *2+ x3 = 0 A x2 + x3 + x4 = 0}. %3D…

A: see below the answer and explanation

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.