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- Calculus Let B={1,x,ex,xex} be a basis for a subspace W of the space of continuous functions, and let Dx be the differential operator on W. Find the matrix for Dx relative to the basis B.arrow_forwardProof Prove that if S1 and S2 are orthogonal subspaces of Rn, then their intersection consists of only the zero vector.arrow_forwardCalculus Repeat Exercise 45 for B={e2x,xe2x,x2e2x}. 45. Calculus Let B={1,x,ex,xex} be a basis for a subspace W of the space of continuous functions, and let Dx be the differential operator on W. Find the matrix for Dx relative to the basis B.arrow_forward
- Repeat Exercise 41 for B={(1,2,2),(1,0,0)} and x=(3,4,4). Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the subspace. a Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B. b Apply the Gram-Schmidt orthonormalization process to transform B into an orthonormal set B. c Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B.arrow_forwardCalculus Let B={1,x,sinx,cosx} be a basis for a subspace W of the space of continuous functions and Dx be the differential operator on W. Find the matrix for Dx relative to the basis B. Find the range and kernel of Dx.arrow_forwardLet A be an mn matrix where mn whose rank is r. a What is the largest value r can be? b How many vectors are in a basis for the row space of A? c How many vectors are in a basis for the column space of A? d Which vector space Rk has the row space as a subspace? e Which vector space Rk has the column space as a subspace?arrow_forward
- About the sets, S1 = {(x,y) ∈ R2: x = y3} S2 = {(x,y,z) ∈ R3: x = (y−z)2} S3 = {(x,y) ∈ R2: x4 + y4=0} S4 = {f(x) ∈ P3(R): f(0) = f(1)} Bearing in mind that P3(R) is the set of polynomials of degree at most 3, with coefficients in R, judge as true or false: (a) Only S2 and S4 are vector subspaces. (b) The sets S1,S2,S3, and S4 are vector subspaces.arrow_forwardLet V=R^2 and let H be the subset of V of all points on the line 2x+3y=6. Is H a subspace of the vector space V? does H contain the zero vector V? is H closed under addition? is H closed under scalar multiplication?arrow_forwardHow do you prove that W= im T when W is a T-invariant subspace, and V=ker + W. Where V is finite dimentional, when you let T be a element V.arrow_forward
- (a) find the orthogonal complement S⊥, and (b) find the direct sum S⊕S⊥. S is the subspace of R3 consisting of the xz-plane.arrow_forwardShow that W={(x1,x2,x3,x4) : x4-x3=x2-x1} is a subspace of R^4 spanned by vectors (1,0,0,-1),(0,1,0,1),(0,0,1,1)arrow_forwardQuestion: Is the subset P(x,y,z) described by 8x-y+2z=0 a subspace of R3? Why or why not? Is my interpretation of subspace correct? Are there any explicit theorems that explain this concept? This is what I came up with: The subset P(x,y,z)described by 8x - y + 2z = 0 is a subspace of R3 because the plane passes through the origin which is used as the positional matrix. Any scalar of x,y, or z will equal 0 and this is an example of a special subspace called the null space. The system is consistent and has infinitely many solutions.arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning