Let V be an n -dimensional vector space and let W C V be an m -dimensional subspace. For each v E V, define Sy = {v+ w: w e W}, and let U = {Sv : v E V}. Define addition in U so that for any x, y E V Sx + Sy = Sx+y and define scalar multiplication so that for any k E R kSx = Skx It can be shown that U is vector space (you do not need to prove this).

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A: To show that H is not a subspace of a vector

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A:

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A:

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Q: Let V be an n -dimensional vector space and let W C V be an m -dimensional subspace. For each v E V,…

A: (c) Given W⊆V with dimW=m≤n=dimV. For each v∈V, define Sv=v+w:w∈W and let U=Sv:v∈V. Let the basis…

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A:

Q: Prove that if S ⊆ R is a vector subspace of R, then either S = {0} or S = R.

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A: The objectives to show the given set is a subspace

Q: is

A:

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A:

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A:

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A:

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A: We have to check

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A:

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A: I have proved the all statements by taking a basis of vector spaces

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A: Solution:Given :     V=ℝ2and      H= (x,y)/ 2x+3y=6   ⊂ ℝ2

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A: Explanation of the answer is as follows

Q: (b) Consider the vector space V = {p(z) = ao + a1r+ azx² | a; € R, deg(p(x)) < 2} of polynomials of…

A: Consider the vector space V=px=a0+a1x+a2x2: ai∈ℝ, degpx≤2 and W=px∈V:p(1)=0

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.