Use Theorem 5.2.1 to determine which of the following are subspaces of p3. 3. (a) all polynomials an +ajx +a2x2 + a3x for which .3 3D0 (b) all polynomials an +ajx + azx + a3x for which ao +a1 +a2+ az =0 .2 .3 (c) all polynomials an +ajx + azx? + azx for which ag, a1, a2, and az are integers (d) all polynomials of the form +ajx, where and a1 are real numbers

Q: Find a basis for the subspace S of R4 consisting of all vectors of the form (a + b, a − b + 2c, b,…

A: We have to find the basis for the subspace S. We have to find the dimension of S.

Q: Find the orthogonal projection of 9. -3 = 4 onto the subspace W of Rª spanned by -3 -4 3 and 6 -3…

A: Solution:-

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

A:

Q: Find a basis for the subspace of R^4 that is spanned by the vectors v1 = (1, 1, 1, 1), v2 = (2,2, 2,…

A: Let us consider set S then the set S is  linearly independent or dependent  Determine the linear…

Q: 5 A subspace W CR' such that W and W- are isomorphic.

A: Given that W is a subspace of ℝ5 let W⊥ is the orthogonal complement of W To verify whether W is…

Q: Find a basis of the subspace of R4 consisting of all vectors of the form ⎡ x1 −8x1+x2…

A: In this we have to find the basis of given vector subspace.

Q: The set of all polynomials of the form ax³ bx² cx + 2, where a,b and can real numbers is a subspace…

A:

Q: 5. Show that if H and H2 are two orthogonal subspaces of H, then Hin H2 = {0}. Does the converse…

A:

Q: Q4(a) Define the null space of an m x n matrix A and prove that it is a subspace of R"

A:

Q: Show that if B is a basis for a subspace H, then each vector in H can be written in only one way as…

A:

Q: let S be the collection of vector [x] (3x1 matrix) in R3 that satisfy the given…

A:

Q: Find a basis for the subspace of R3 spanned by s. S = {(1, 5, 9), (-1, 6, 9), (2, 6, 1)}

A:

Q: Find a basis for the subspace of R³ spanned by S. S = {(1, 2, 6), (-1, 3, 6), (2, 3, 1)}

A: The given subspace is: S= (1, 2, 6), (-1, 3, 6), (2, 3, 1)

Q: Let P2 be the vector space of all polynomials of degree 2 or less, and let H be the subspace spanned…

A:

Q: Suppose V is an inner product space with dim V = 7 and U is a subspace of V with dimU = 2. Then dim…

A: We know that  The sum of the Dimensions of a subspace and the dimensions of its orthogonal…

Q: 2.14. Prove that two subspaces V1 and V2 are linearly independent if and only if Vịn V2 = {0}.

A: V1 and V2 are two subspace.

Q: 1. Compute the orthogonal complement of each of the following subspace of the given vecto (a) W =…

A: We have to find orthogonal complement of given subspace: Concept: If x is orthogonal vector to…

Q: If W is a subspace of V and B is a hasis for V, then some subset of B is a basis for W. Select one:…

A:

Q: The set of all 3 x 3 matrices of the form 0 0 0 0 0 -5x 0 0 0 where x is a real number, is a…

A: Given :M=00000000-5x

Q: Find a basis for the subspace of R3 spanned by S.S = {(4, 4, 8), (1, 1, 2), (1, 1, 1)}

A:

Q: Find the orthogonal projection of v = projv (v) = [B] 9 onto the subspace V of R³ spanned by 4 and…

A:

Q: If a subspace of V of R contains the standard vectors ei, e2, eg, then V must be R. True False

A: We will show that V is a subset of R³. Then, we can claim that, V = R³

Q: Find the orthogonal projection of 5 -19 -6 onto the subspace W of R* spanned by 4 4 and 1 -2 3 projw…

A:

Q: Determine whether the following are subspaces of C[−1, 1]: The set of odd functions in C[−1, 1]

A: Consider the given set: C[−1, 1] Here the objective is to prove that the set of odd functions in…

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

A:

Q: Criticize the following argument: For any vector v, we have Ov = 0. So the first criterion for…

A: Let W be the subset of a vector space V. If W is itself a vector space under the operations of V,…

Q: 2) Find all the invariant subspaces of the following matrix 0. 1 0 1 1 1

A: Given matrix is, Let, A=010010111 Now, calculating the eigen value of the given matrix.…

Q: If W is a subspace, then ||projwv||² + ||v – projwv||2 = ||v|?.

A: Given W is a subspace of a vector space V. Now projWv denotes the projection of vector v onto the…

Q: Find a basis for the following subspace of Rª over R: 3 1 4 7 span 10 11 13 What is the dimension of…

A:

Q: Are the following statements true or false? 1. If W is a subspace of R" and if v is in both W and W,…

A:

Q: Find a basis for the subspace of R3 spanned by S. S = {(5, 9, 9), (1, 2, 2), (1, 1, 1)}

A: To find the basis for the subspace of R3 spanned by the given set of vectors S.

Q: -11 -3 1 2 Find the orthogonal projection of v = onto the subspace W spanned by 43 -5 -3 3 4 5 projw…

A: Given: v→=-11243-3 and W is a subspace spanned by -32-53, -3254, 12-35.

Q: The set {(x,3): x is in R)} is a subspace of R2

A: The given set is x,3: x is in R. It is known that any subspace should contain identity element.

Q: 7. The set of all 2 x 2 upper triangular matrices is a subspace of M22- a. True. b. False.

A: Use two step test of subspace to solve this problem.

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

A: False

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

A:

Q: Let 5 Find a basis of the subspace of R4 consisting of all vectors perpendicular to v.

A:

Q: The set of all 3 x 3 matrices of the form 1 0 0 0 1 0 00 -a Select one: True O False + 13x3 where a…

A:

Q: 2. Determine which of the following is a subspace of R". (a) {(x1,x2,...X) 1 x = 0} (b) {(x1,…

A:

Q: Determine whether W = {ao + a1x + a2x² + a3x°|ao + a2 = az ɑ1} is a subspace of P3.

A:

Q: (2) Find a basis for the subspace W%3D { (а+c, 2a — b, За — 2b — с, b + 2с) | а, b, с € R} of R4.

A:

Q: Let V = M22 and W = {invertible 2 x 2 matrices}. Determine whether W is a subspace of M2. %3D

A:

Q: x1 + x2 + x3 = 0 x4+x₂+2x3 = 0

A:

Q: Find the orthogonal projection of -13] 15 onto the subspace W of R4 spanned by 4 and 5 -2 -2…

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: Find the orthogonal projection of [-10] = -15 12 onto the subspace W of IR spanned by 日-日 1 9. and 6…

A:

Q: The following subset W of R² where W is the set of all vectors of the form CH with the usual…

A:

Q: Find a basis for the subspace of R³ spanned by S. S = {(1, 5, 7), (-1, 6, 7), (2, 6, 1)}

A: Write the given set of elements as the rows  to form a 3x3 matrix. The space spanned by the rows of…

Q: -3 -10 -2 -2 and let W the subspace of R4 spanned by ű and v. Find a basis of W+, the orthogonal…

A: