Consider the subspace of P2 given by V = span ({-1+r+x²,1– 2r +r, -1+ 3r, 2 – 3r}). Answer the following questions: a. Determine a basis for V.

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: Find a basis for the subspace of P3 consisting of all vec- tors of the form at 3 + bt 2 + ct + d,…

A: Given: The vectors are of the form: at3+bt2+ct+d Also given that: c=a-2db=5a+3d Putting the given…

Q: Show that W is a subspace of R¹.

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Q: Determine if the set W of vectors (x1 , x2) in R such that xr? + x2 = 1 is a subspace of R’.

A: W is not a subspace of R²

Q: Let W, and W2 be subspaces of a vector space V. Prove that V is the direct sum of W, and W2 if and…

A: First assume that V is the direct sum of W1 and W2  W1+W2=V and W1∩W2=0 We have to prove that each…

Q: Verify that W is a subspace of V. In each case, assume that V has the standard operations.W = {(x1,…

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Q: verify that W is a subspace of V. In each case, assume that V has the standard operations. 1. W =…

A: Given: V=ℝ4 and W=x1, x2, x3, 0|x1, x2, x3∈ℝ

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 –…

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Q: In each part, find a basis for the given subspace of R4 .(2) All vectors of the form (a, b, c, d)…

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Q: Let Z E R³ | y Prove that Z is a subspace of R³. = Z

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Q: Consider the vectör X1 = X2 = 4. in R3, and let W be the subspace Span{x1, X2}. (a) Find an…

A: Given :-  Consider the vectors  x1 = 2-11   , x2 = 5-44 in ℝ3. and let W be the subspace span  x1 ,…

Q: If S is a subspace of R", then so is the orthogonal complement of S.

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Q: Let W {(a,b, c, d)| 5a - c = 0 } be a subspace of R*. Then a basis for W is

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Q: Find a basis and the dimension of the given subspace of R4: a + c a – b b+c E R4 -a + b

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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: Let vị 2,v2 8,and w Determine if w is in the subspace of R generated by v1 and v2.

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Q: Determine if the set W of vectors (X1 , X2) in R such that x1² + x2 = 0 is a subspace of R?.

A: We have to determine whether W is a subspace of R^2 or not.

Q: Prove that W1 = {(a,b) E R² : 4a – b= 0} is a subspace of R2.

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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:

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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: Find a basis for subspace S of M2,2. s ={ [ ] : 4a – b = 2c + 2d ; a, b, c, d E R - C

A: We have to find basis of subspace S of M2x2 matrix.

Q: Find a basis of the subspace of R" that consists of all vectors perpendicular to both 1 and 1 -5 5…

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Q: Let w be the subset of R' consisting of all vectors of the form b where b=2a+1. Examine W is a…

A: Consider a vector space V(F) then a subset W of V is said to be a subspace of V if   αA+B∈W…

Q: Let W be the subspace spanned by the given vectors. Find a basis for W-. 1 -1 1 W1 W2 -5 -2 1

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Q: Let 1 -3 and let W the subspace of R' spanned by u and v. Find a basis of W-, the orthogonal…

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Q: Consider the set of vectors in R such that 1223 0. If these form a subspace, show that is satisfies…

A: See below for solution.

Q: Determine the dimension of the subspace of R consisting of all XeR* such that x1 + 2x, = 0 and X3 –…

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Q: let H= { a b 0 \ a ∈ R and b∈ R.} Show that H is a subspce of R3 by showing that it…

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Q: Verify that W is a subspace of V. In each case, assume that V has the standard operations.{(x, y, 4x…

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

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Q: Let W be a subset of the vectors in R where W = X2 X4 -x3 = x, - x, }. Show that W is a X3 3. X4…

A: We will use the formal definition of a subspace of a vector space to prove that, W is a subspace of…

Q: Let M = x2 :¤1, x2 E R X1

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Q: For the subspace a H= b :a-5b+4c=0 d a basis is 1 O True O False

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Q: If W is a subspace of ℝ4 whose dimension is 2 then is any set of 2 vector is a basis for W

A: Answer

Q: Let W be a subspace of R". A set of vectors {v1, ..., Úp} in W is defined to be a basis of W if {v1,…

A: Let ℝn be a vector space. Let W be a subspace of ℝn. Then,  is W={v1→,......,vp→}a basis. Definition…

Q: Show that the given set V is closed under addition and multiplication by scalars and is therefore a…

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Q: Find a basis and the dimension of each of the following subspaces of R" a. W1 = v: u + v = 0 and u…

A: Given :  The subspaces of  ℝ3  are given as ; a)   W1 = { (u, v, w) : u+v=0  and u-2w=0 } b)   W2 =…

Q: Let A be an n x n matrix. Verify that V = {x e R" : Ax = 3x} is a subspace of R".

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Q: If V is any three-dimensional subspace of R°, then V has infinitely many bases.

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Q: Consider a subspace and a vector (:) (3) 1 H = Span- 2 X = %3D 2 7 Determine if x e H. If so find…

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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,

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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: Determine: if W = the set of all vectors in R2 whose components are integers is a subspace.

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Q: Let W be the subspace spanned by the given vectors. Find a basis for Wt. W = W2 = -5

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

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Q: Let W1 and W2 be subspaces of a vector space V. Prove that W1 ∪W2 is a subspace of V if and only if…

A: It is given that W1 and W2 are the two subspaces of the vector space V.

Q: Let {w1, w2, ..., Wk} be a basis for a subspace W of the vector space V. Let v be a vector in W1.…

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Q: Find a) basis for each of the following subspaces of the given vector space V and b) dimension of W.…

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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.