Let vi # v2 E R" and define W1 Span{v2}. Use the definition to show W+n W, is a subspace of R". Span{v1} and W2

Q: Prove thatII = {(x, y, z)|x – 2y+ 3z = 0}is a subspace of JR³, %3D -

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Q: Let R be the field of real numbers. Then {(x,x,x): x ER}is a subspace of R (R) ?

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Q: Show that W is a subspace of R¹.

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Q: Let W = {a + bx + cx² + dx³[ c – 3d = 0 } be a subspace of P3. Then the dimension of W is equal to O…

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Q: 2. Find the closest point to y in the subspace W spanned by vị and v2. 3 1 -2 y V2 13 3

A: To find  The closest point y in the subspace W spanned by v1 and v2.

Q: _. Let u = 6 " and let W be the set of all x in R3 such that u x = 0. (a) Show that W is a subspace…

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Q: Let H = Span{₁,..., If u € R¹ satisfies vú = v½ u then u belongs to H¹. p} be a subspace of R" . =…

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Q: (c) Find a subspace U such that R³ = W₁ OU.

A: C. Hint: Use the result of part (b) in the part (c) and obtain the required result.

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

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Q: Let W = {a + bx + cx² + dx°|c - 3d = 0} be a subspace of Pa. Then the dimension of W is equal to 3.…

A: PnF is set of all polynomials of degree nand smaller . PnF denotes the vector space over field F…

Q: Let V = {(x, y, 2x – 3y)|x, yɛR}. Prove that V is closed for addition and scalar multiplication i.e.…

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Q: Find the closest point to y in the subspace W spanned by v1 and v2 1 ,V2 = 1 y = V1 = -3. I • E 2 b)…

A: Here we have to find the closest point to y in the subspace W spanned by v1 and v2 .

Q: a. . Is the set of vectors of the form a subspace of R³? Show your proof. La+2b

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Q: Find the orthogonal projection ŷ of y = onto the subspace W = Span { uị u2 5 Ex: 1.23 : ŷ =

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Q: Let S be the subspace of R2 spanned by e1 and let T be the subspace of R2 spanned by e2. Is S ∪ T a…

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Q: Q.4 Show that U {(x, -x) |x in R} is a subspace of R. %3D

A: To Prove U is a subspace of vector space V 1) 0∈U  If x, y∈U  and a∈V  2) Closed under vector…

Q: Let X =R, and M = (0} , the M is not subspace of R O True O False

A: Subspace of a vector space

Q: Suppose V is finite-dimensional and U is a subspace of V such that dim U = dim V. Prove that U = V.…

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Q: Suppose V is finite-dimensional and U and W are subspaces of V with W^0 ⊂ U^0. Prove that U ⊂ W.

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Q: a Is the set of vectors of the form a subspace of R³? Show your proof. La+2b

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Q: Verify that V5 is a subspace of RS

A: Please check step 2 for solution.!

Q: Let W = :a +5c = 0} be a subspace of M22. %3D Then dimension of W is equal to: None of the mentioned…

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Q: 4. Show/prove whether the following is a subspace. {x € R3:x =| 3 +s 2+t| 1 ,for s,t e R} 2

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Q: a. Is the set of vectors of the form a subspace of R3? Show your proof. La+2b|

A: Since you have asked multiple questions, we will solve the First Question for you. If youwant any…

Q: Suppose that X = ER' :z + y + z = 0 and ² + y² = z² Determine (with justification) whether X is a…

A: Given  X=xyz∈ℝ3:x+y+z=0 and x2+y2=z2

Q: Show that the set S = {(x, y) in R2 such that x =3y } is a subspace of the vector space R2 NOTE!…

A: That's easy. Have a good day!!!

Q: Suppose U and W are two-dimensional subspaces of R3. Show that U∩W≠{0}

A: We want to show that  U∩W≠{0}

Q: Let W = {a + bx + cx2 + dx'|c-3d = 0} be a subspace of P, Then the dimension of W is equal to %3D 4.…

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Q: Find the orthogonal projection of V = nto the subspace W of R* spanned by Uj = U2 = Uz = -6/4 -6/4 w…

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Q: What is the dimension of w = {A € M2x2 (R) : A"| =ö} ĄT -2 1 2 This is not a subspace

A: W=AEM2×2(R) : AT1-2=→0  Let A=abcd, a,b,c,dER     AT=acbd     AT1-2=→0  ⇒acbd 1-2 =00 ⇒a-2cb-2d=00…

Q: Let W = {a + bx + cx? + dx³|c – 3d = 0} be a subspace of P3. Then the dimension of W is equal to…

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Q: Consider the vectors v1, v2, ..., Um in R". Is span (01,..., īm) necessarily a subspace of R" ?…

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Q: Give a counter example to show that W is not subspace of R³. W=set of (x, y, z) where x + y = 1

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Q: Suppose that U cr|xyer F4 y EF, U is a subspace of F4- you don't have to! Find a subspace V of F4…

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Q: ind the closest point to y in the subspace W spanned by vị and v2 2 y =| 1 |,v1 = ,V2 = |-2 2 а) b)…

A: We will use the following method.

Q: Is M = {(x,y,z)|x is nonnegative} a subspace of R³? Yes or No? Justify or give/find a…

A: Given  M =  x, y, z  x is nonnegative

Q: 3) Explain why S, = {(x, y) E R²:y > 0} and S1 = {(x, x²): x E R} are not subspaces of R?. %3D

A: We will solve both these parts using only basic knowledge of subspaces and vector spaces (in clear…

Q: . Let u = 6 6 " and let W be the set of all x in R3 such that u x = 0. (a) Show that W is a subspace…

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Q: Give a counter example to show that W is not subspace of R³. W=set of (x, y, z) where x + y = 1 6. ,

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Q: Q. Show that (a,b,c) where c=a+b is subspace of R

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Q: Suppose that X = eR' : 2 + y + z = 0 and r²+y² = z² } . Determine (with justification) whether X is…

A: We have to check whether the set X is a subspace of the vector space ℝ3. We have to give a…

Q: Let W = {(" ):a = 2c = b = –3d} be a subspace of M22. Then dimension of W is equal to: None of the…

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Q: Find the closest point to y in the subspace W spanned by v1 and v2 -21 1 y = | 2 = 2,v2 l1. -1 1 0.…

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Q: ver the field of reals is a subspace of the specified vector space | A = V = E R"Xm a2 ат

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Q: Suppose V is finite-dimensional and U is a subspace of V such that dim U = dim V. Prove that U = V.

A: A set B is a basis of a vector space V if its elements are linearly independent and every elements…

Q: For a certain subspace W, the orthogonal projection of the vector y = 5 onto W 4 is ŷ = |1. Use this…

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Q: Let U and WW denote subspaces of R", and assume that U C W. If dim U = n - 1, show that either W =…

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Q: Find the orthogonal projection ŷ of y = 3 onto the subspace W = Span { u1 = , U2 = 3 Ex: 1.23 %3D

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Q: {{} } 5. Let W be the subspace of R³ spanned by write j= |3 as the sum of a vector in W and a vector…

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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.