Consider the subspace W = {p € P2(R) : p(4) = 0} of P2(R). Prove that B = {x – 4, (x – 4)²} is a basis for W.

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

A: We need to find the basis for the given subspace:

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

A:

Q: If W is an invariant subspace for T then W is invariant under every polynomial in T

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: 5. Let W be the set of all vectors p(t) in P, such that p(0) = 0. Show that W is a subspace of P,…

A:

Q: Which of the following is a basis for the subspace {:)- x y z W = x = y + z and x, y,z E R}C M2x3(R)…

A: We have to satisfy the given condition in the matrix i.e. x = y+z

Q: Find a basis and the dimensi on of the subspace W of P(t) spanned by (a) u=t+ 28 – 21+1, v= P + 32 –…

A:

Q: 4. Assume that T E L(V) and that U1 and U2 are T-invariant subspaces of V. Prove that U1 n U2 is…

A:

Q: Let P3[t] denote the vector space of all polynomials in t of degree at most 3 with real…

A: To show that the given set is a basis for P3[t], it is sufficient to show that any element in P3[t]…

Q: Find a basis for the subspace а : a, b, c, d E R with 3a + b=0 and c+ d = 0 C W = 6. 6. d of R4.

A: Since 3a+b=0 Therefore b= -3a  And since  c+d=0  Therefore d= -c

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: 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: Let W = {a + bx + cx? + dx³ € Pg : c = a – b+ 2d} be a subspace of P3. Then a basis for W is:

A:

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: 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: The given vectors form a basis for a subspace W of R3 or R4. Apply the Gram-Schmidt Process to…

A:

Q: If W is a subspace of R" of dimension k with a basis {v1,e,2,..., v%} then W Range(|v1 v2 vn)). ...…

A: Given: Let W be a subspace of Rn and dimk=k Let B be a basis of k then B=k Also B is a linearly…

Q: Consider the following vector v ER and the subspace SCR¹. -0-000 S= = span V= V₂ (a) Find an…

A: consider the following vector v∈ℝ4 and the subspace S⊂ℝ4 where v=1234 and S=span1111, 01-11, 0110…

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: 2. Consider the subspace im(A) of Rº, where 1 1 A = | 1 1 3 Find a basis of ker(A"), and draw a…

A: Given matrix A=111213 AT=111123

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

A:

Q: 1. Suppose that V is a vector space and H is a subspace of V and the vectors vl,v2,...,vj all belong…

A: We have given that V is a vector space and H is a subspace of V, so H is also a vector space . Now…

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: Let 3 u= and let W the subspace of Rª spanned by ū and v. Find a basis of W+, the orthogonal…

A: Given vectors, u→=-3-10-2-2,v→=132-3 The objective is to find the basis of W⊥ the orthogonal…

Q: Find a basis for the subspace 5а + 8b 7b E R2x2 a, b € R}of R2x2. W За 26E Ex: 1 7 A basis for W is…

A:

Q: Let T be a function from P3 (F), the vector space of polynomials with degree at most 3, to P5 (F),…

A: Let, T be a function from ℙ3(F), the vector space of polynomials with degree at most 3, to ℙ5(F)…

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

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Q: Let P be the projection matrix corresponding to a subspace S of Rm. Show that P2 = P

A: Given :Let p be the projection matrix corresponding to a subspace             S of RM

Q: 2. Show that {1,x – 1, x (x + 1)} is a basis of the space of polynomials P2 (IR) and that W = {p(x)…

A: If the elements of the set are linearly independent then the set is a basis set. If Wvis a subspace,…

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

A:

Q: Given a basis B = {V1, V2,, Vn} of Subspace S, there is a unique way to express any ... vE S as a s…

A: We will first write 2 expressions of v as a linear combination of basis vectors in B. Then, we will…

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.

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Q: Find a basis and the dimension of the subspace W = {(a+1)x+(2- )x²; a,b e R} of P,

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: Find a basis for the subspace W p(x) E P2 p(5) = -p(-5)} of P2. A basis for W is { x2 -Ex: 1 , x

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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: An m×n matrix A is called upper triangular if all entries lying below the diagonal entries are zero,…

A: Proof:

Q: Let A be an m×n matrix. Find all b such that the set of solutions to Ax = b is a subspace and prove.

A: Given: Let A be an m×n matrix.  The objective is to find that the set of solutions to Ax =b is a…

Q: Find a basis and the dimensi on of the subspace W of V = M22 spanned by A B

A: V=M2X2R SPANNED BY A=1-5-42, B=11-15, C=2-4-57, D=1-7-51 Solution:- To find: A basis and the…

Q: Let U={p E p2 : p(2)=0}.Show that U is a subspace of p2 and find a basis for U.

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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: 1 1 11. Find a basis for the subspace of R4 spanned by u1 = U2 = U3 and -3 1 1 1 U4 =

A:

Q: Find a basis and the dimension of the subspace W = {(a+1)x+(2-b)x²; a,b e R} of P,.

A:

Q: ) Let W C R³ be a subspace with basis {{} 3 -5 -1 1 Use Gram-Schmidt process to produce an…

A:

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:

Q: Let W be the subspace of P2 defined by all p(x) in P2 such that p(x) = 0. (a) Construct a basis for…

A: (a) A polynomial p(x)=ax2+bx+c satisfies p(1)=0. Hence every p∈W is of the form,…

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.