If W, and W2 are finite - dimensional subspaces of a vector space V, then W + W2 is finite – demesional and dim W, + dim W2 = dim (W1 n W2) + dim (W1 + W2).

Q: Find the dimension of the vector space U of all linear transformations of V into W for each of the…

A: a)- Given: V=R2 , W=R3 Since R2 is 2-dimensional vector space and  R3 is 3-dimensional vector space,…

Q: Suppose U, W,, W, are subspaces of a vector space V. Show that (Un W,) + (UnW,)Cun (W, + W,) Find…

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Q: If V3(R) is a vector space and W₁ = {(a,0, c): a, c = R} and W₂ = {(0,b,c): b, c = R} are two…

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

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Q: O Linear subspace L is given by the equations 2r1 + r2 + 3r3 - 14 = 0 = 0 3r1 + 2r2 – 2r4 3r1 + 12 +…

A: Let AX=0 213-1320-2319-1x1x2x3x4=000R2→R2-32R1 , R3→R3-32R1…

Q: 6. Let Vi and V2 be subspaces of R" such that Vi C V2. Show that any vector v E R" can be uniquely…

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Q: If W₁ and W₂ are subspace of a vector space V(F), then show that W₁ + W₂ is also a subspace of V(F).

A: W1 and W2 are the subspace of a vector space BF We have to show W1+W2 is also a subspace of BF

Q: V is a 3-dimensional vector space with basis By (V1, V2, V3) and W is a 3-dimensional vector space…

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Q: If v1 and v2 are in R" and H=Gen{v1, v2}, then H is a subspace of R" Justify

A: Given that : and are in  Let Now, we  have to recall  the definition of subspace of a…

Q: f W1 and W2 are subspace of a vector space V(F),  then show that W1+W2 is also a subspace of V(F)

A: Given that W1 and W2 are two subspace of a vector space V over a field F.

Q: Please go through each option with explanation

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Q: The vector x is in a subspace H with a basis B = {b₁,b₂). Find the B-coordinate vector of x. 2 b₁ =…

A: We are given that           b1 = (2, -3)t          b2 = (-1, 5)t        Now, x is in H with the…

Q: Q: Let W, ={(a, B.Y, 8): a=y+28, y= 28} and W2 {(a, B.Y, 8):a-y-0} be two subspaces of V,(R). Find a…

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Q: Let S = {a+bx+cx²: = {V1, V2, V3} and T a, b, c = R}, where v₁ = 1 + x + 3x², v₂ = −2 − 2x – 7x², v3…

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Q: 2. Let P3(R) be the vector space of all polynomials of degree 3 or less (and the zero polynomial).…

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Q: If V is a linear subspace of R", and x1 , ×2 , ... , Xn E V, then we must have span {x1, X2, Xn} =V.…

A: To determine that the statement, if V is a linear subspace of ℝn and x1, x2,....,xn∈V, then spanx1,…

Q: 2. Let M2x2 (R) be the set of all 2 x 2 matrices (with real number entries). That is a b {[%] | a,…

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Q: If V3(R) is a vector space and W₁ = {(a,0, c): a, c = R} and W₂ = {(0, b, c): b, c = R} are two…

A: Provided that V3 is three dimensional vector space over field R and V has two W1 and W2 subspace…

Q: . Consider M₂ (R) as a vector space and its subspaces U₁= {A € M₂(R): A¹ = A} and U₂ = {A € M₂(R) :…

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Q: If scalar multiplication and addition is redefined on R² in the following way, show that it is NOT a…

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Q: be the vector space of all polynomials of degree 3 or less in the variable z. Let PI(T) 2+x+1²,…

A: In this question, we check the given set forms a basis or not. Then we use elementary transformation…

Q: If we know that W is a subset of the vector space V and we know that W contains the zero vector 0,…

A: This statement is False.  By using property of subspace we can prove this statement is False.

Q: Let B= = : {b₁, b₂} and C = {c₁, c₂} be bases for a vector space V, and suppose V = b₁ b2 = [v] =…

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Q: Prove that if U and V are subspaces of R" and U <V then V+<U+.

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Q: Let Uj and U2 be vector subspaces of a vector space V over the real numbers. Suppose that v1, V2, V3…

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Q: Suppose W is the subspace of R“ satisfying the equation x1 + x2 + x3 + x4 : A. Find a basis for W-.…

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Q: Let W be a subspace spanned by the u's, and write y as the sum of a vector in W and a vector…

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Q: V 3 (R) is a vector space and W₁ = {(a, 0, c) : a, c = R} and W 2 = {(0, b, c) : b, c € are two…

A: Provided that V3 is three dimensional vector space over field Rand V has two W1 and W2 subspace…

Q: Let V be a finite dimensional vector space over F and let W₁ and W₂ be two subspaces of V. Then,…

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Q: Show that V = W₁ © W₂ for the given vector space V and the given subspaces W₁ and W₂. (a) V = R³, W₁…

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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 the subspace spanned by u₁ and u2, and write y as the sum of a vector in W and a vector…

A: Let W be the Subspace Spanned by u1 and u2,Write y as the sum of a vector in W and a vector…

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

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Q: 3 Let M₂(R) be the vector space of all 2 × 2 matrices with entries from R. Show that M₂ (R) = W₁ W₂,…

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