Answered: Problem 1 Find a basis for the… | bartleby

Elementary Linear Algebra (MindTap Course List)

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CR: Review Exercises

Problem 47CR: Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}

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Problem 1 Find a basis for the following subspace of the vector space of polynomials of degree
< 3:
W := span {1+x+x²+x³, −1+5x+x² +11x³, −1+2x+5x³, 1+ 6x + 4x² + x³}

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Find a basis for R2 that includes the vector (2,2).
Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}
Suppose y1 ( x), y2 ( x), y3 ( x) are three different functions of x. The vector space they span could have dimension 1, 2, or 3. Give an example of y1, y2, y3 to show each possibility.
2. (a) Write one of the four vectors 1 = 2 8. as a linear combination of some or all of the others. (b) Find a linearly independent subset of {u, v, w, a} that has the same span (in other words, find a basis for the subspace spanned by these vectors).
This is a basis and dimension problem from the vector space section for linear algebra
,x43x-2 4. Determine whether the veltars X-I and X²-3x+2 dre linearly independent Or dependent in P2 (te vector Space of Polynomials of doyree less than equal to 2). or
Find a basis for the subspace of R4 determined by the equation 2x-3y+4z-w=0
2. (a) Write one of the four vectors as a linear combination of some or all of the others. (b) Find a linearly independent subset of (u, , u, 7} that has the same span (in other words, find a basis for the subspace spanned by these vectors).
Let S be the subspace of R¹ consisting of the solutions to the following system of equations: x1-2x2-2x3-3x4 -2x1+7x2+10x3-6x4 -2x1+5x2+7x3-2x4 = 0 = 0 = 0 Give a basis for S. Number of Vectors: 1 {[]}
Find a basis of the subspace of R* consisting of all vectors of the form x1 -9x1+ x2 -9x1– 2x2 -6x1+4x2] Your answer should be a list of row vectors separated by commas. (Click vector to learn about entering vectors.) Basis = {
The set of all polynomials of degree 6 under the standard addition and scalar multiplication operations is not a vector space because" We can find a polynomial P(x) for which 1-P(x)=P(x) We can find two polynomials P(x) and Q(x) for which P(x)-Q(x)=Q(x)•P(x) It is not closed under addition. We can find a polynomial P(x) such that (c+d)P(x)zcP(x)+dP(x).
Let W be the subspace of the vector space of polynomials of degree 3 spanned by 2₁ = 1 + 3x - 2x² + x³, v₂ = -3-9x + 6x²-3x³, v3 = x + x². x+x² - 2x³, 24 = 1+ 2x - 3x² + 3x³, vs = -2-5x + 5x2 - 4x3. Which of the following statements are true? (i) The set Bw = {₁, 23,1₁) is a basis for W. (ii) The set Cw = {₁, 4) is a basis for W. (iii) The set By = {1, 23, 1, 2} is a basis for the space of polynomials of degree 3. (iv) The set Cy (v1,vs, 1,22} is a basis for the space of polynomials of (v) a +bx+cx² + da3 is in W, if 5a - b + d = 0 and 7a+2b+c=0 (vi) 1-x+x²-23 EW and -1-4x+x² + x³ € W. degree 3. A) () and (v) B) (i), (iii) and (iv) C) (i), (ii), and (vi) OD) (ii) and (iv) E) (), (iv) and (v)

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