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Let T : R^n rightarrow R^m be a linear transformation. Given a subspace Z of R^m, let 5 {v
elementof R^n | T(v) elementof Z). Show that S is a subspace of R^n.
Solution
Let v,w in S
Then T(v) and T(w) belongs to Z
Consider av+bw , a,b are scalars
We need to see if av+bw are in S
For this consider T(av+bw) = aT(v)+bT(w) (because T is linear transformtaion)
Now T(v) and T(w) are in Z
And so aT(v) and bT(w) are in Z
This => aT(v)+bT(w) are in Z
Hence we get that T(av+bw) is in Z
So we get av+bw is in S
Hence S is subspace of Rn

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Let T R^n rightarrow R^m be a linear transformation. Given a subspa.pdf

  • 1. Let T : R^n rightarrow R^m be a linear transformation. Given a subspace Z of R^m, let 5 {v elementof R^n | T(v) elementof Z). Show that S is a subspace of R^n. Solution Let v,w in S Then T(v) and T(w) belongs to Z Consider av+bw , a,b are scalars We need to see if av+bw are in S For this consider T(av+bw) = aT(v)+bT(w) (because T is linear transformtaion) Now T(v) and T(w) are in Z And so aT(v) and bT(w) are in Z This => aT(v)+bT(w) are in Z Hence we get that T(av+bw) is in Z So we get av+bw is in S Hence S is subspace of Rn