Let T be a linear transformation from R2 into R2 such that T(1,0)=(1,1) and T(0,1)=(−1,1). Find T(6,1) and T(−5,1) T(6,1)=? T(−5,1)=?

Question

Let T be a linear transformation from R2 into R2 such that T(1,0)=(1,1) and T(0,1)=(−1,1). Find T(6,1) and T(−5,1)  T(6,1)=?  T(−5,1)=?

meldElafellrbo · Accepted Answer

Concept: property of linear transformation from Rn to Rm states that T(v1+v2)=T(v1)+T(v2) where T is linear transformation and v1, v2 are vectors from Rn   T(a⋅V1)=aT(V1) where a is scalar, V1 is vector     given T(1,0)=(1,1)   T(0,1)=(−1,1)   T(6,1)=? and T(−5,1)=?    T(6,1)=T(6+0,0+1)=T((6,0)+(0,1))   using properties of linear transformation we get    =T(6,0)+T(0,1)  =T(6(1,0))+T(0,1)   =6T(1,0)+T(0,1)   =6(1,1)+(−1,1)   =(6,6)+(−1,1)   =(6−1,6+1)=(5,7)   linear transformation of (6,1) is (5,7)   T(−5,1)=T(−5+0,0+1)=T((−5,0)+(0,1))    using properties of linear transformation we get    =T(−5,0)+T(0,1)   =T(−5(1,0))+T(0,1)   =−5T(1,0)+T(0,1)   =−5(1,1)+(−1,1)   =(−5,−5)+(−1,1)    =(−5−1,−5+1)=(−6,−4)   linear transformation of (-5,1) is (-6,-4)

Let T be a linear transformation from R^{2}\ \text{into}\ R^{2}

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