Let T:R3→R3 be the linear transformation such that T[111]=[−25−2],T[110]=[414],T[100]=[1−11] a) Find a matrix A such that T(x)=Ax for every x∈R3 b) Find a linearly independent set of vectors in R3 that spans the range of T

Question

Let T:R3→R3 be the linear transformation such that  T[111]=[−25−2],T[110]=[414],T[100]=[1−11]  a) Find a matrix A such that T(x)=Ax for every x∈R3   b) Find a linearly independent set of vectors in R3 that spans the range of T

gekraamdbk · Accepted Answer

Step 1  Presumably your book wants the answer in terms of the standard basis. For ease of typing, I will use row vectors instead of column vectors.  Due to linearity, we can see:  T(0,0,1)=T(1,1,1)−T(1,1,0)=(−2,5,−2)−(4,1,4)=(−6,4,−6)  Similarly, we can see  T(0,1,0)=T(1,1,0)−T(1,0,0)=(4,1,4)  So theh  −(1,−1,1)=(3,2,3)  matrix for T with respect to the standard basis is:  [13−6−12413−6].  The range of this transformation will be spanned by the column vectors of this matrix. We must find a linearly independent subset of them in order to answer your question.

Micheal Hensley · Accepted Answer

Step 1  You should be able to see that the set of vectors  u1=[111],u2=[110],u3=[100]  spans 𝟛R3. Let 𝟛X∈R3 so we can write it as  X=a,u1+b,u2+c,u3.  where a,b,c are the coordinates of the vector X with respect the above basis. Apply the operator T to the vector X gives  T(X)=aT(u1)+bT(u2)+cT(u3)  =a[−25−2]+b[414]+c[1−11]  =[−24151−1−241][abc]=AX.

Let T:R^{3}\rightarrow R^{3} be the linear transformation such that \(T\left[\begin{array}{c}1\\1\\1\\ \end{array}\right]=\left[\begin{array}{c}-2\\5\\-2\\

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