Please explain your steps a Construct a ndimensional subspac

(a) Construct a n-dimensional subspace W of each vector space. (b) Prove that W is a subspace and n-dimensional. 2-dimensional subspace of R^4. 2- dimensional subspace of M_2 times 3

Solution

(a) Which vector space? For Rⁿ, it is span { e₁,e₂,…e_n} where e = (1,0,0,…0)^T e₂ = (0,1,0,…,0)^T,…, e_n = ( 0,0,0,…,0)^T.

(b) Subspace of which vector space?

(c ) A 2 –dimensional subspace of R⁴ is W = span{(1,0,0,0)^T, (0,1,0,0)^T} . Any 2 arbitrary vectors of W are of the form X = (a,b,0,0)^T and Y = (p,q,0,0)^T where a,b,p,q are arbitrary real numbers. Let c be an arbitrary scalar.Then X+Y=(a,b,0,0)^T+(p,q,0,0)^T=(a+p,b+q,0,0)^T=(a+p)(1,0,0,0)^T+(b+q)(0,1,0,0)^T belongs to W. Further cX = c(a,b,0,0)^T = (ac, bc,0,0)^T = ac(1,0,0,0)^T+bc(0,1,0,0)^T belongs to W. Therefore, W is a vector space, and hence a subspace of R⁴.

(d ).W = span {e₁,e₂} is a 2 dimensional subspace of M_2x3 where e₁ =

1

0

0

0

0

0

and e₂ =

0

1

0

0

0

0

Let X = ae₁ +be₂ and Y = pe₁+qe₂, where a,b,p,q are arbitrary real numbers. Let c be an arbitrary scalar. Then X+Y = ae₁ +be₂+ pe₁+qe₂= (a+p)e₁ +(b+q)e₂ belongs to W. Further cX = c(ae₁ +be₂) = (ace₁ +bce₂) belongs to W. Therefore, W is a vector space, and hence a subspace of M_2x3.

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