Answer to Question #219575 in Linear Algebra for danny

Question #219575

find a linear transformation T:R^3->R^3 whose image is spanned by (1,2,3 ) and (4,5,6)


1
Expert's answer
2021-07-22T10:14:56-0400

Since "\\begin{pmatrix}\n 1 \\\\\n 2 \\\\\n 3\\\\\n\\end{pmatrix}" and "\\begin{pmatrix}\n 4\\\\\n 5\\\\6\\\\\n\\end{pmatrix}" spans "Img(T)," then for any "y \\in Img(T), y" can be written as a linear combination of "\\begin{pmatrix}\n 1 \\\\\n 2 \\\\\n 3\\\\\n\\end{pmatrix}" and "\\begin{pmatrix}\n 4\\\\\n 5\\\\6\\\\\n\\end{pmatrix}" i.e.


"T(x)=y=\\begin{pmatrix}\n 1 \\\\\n 2 \\\\\n 3\\\\\n\\end{pmatrix}\\alpha + \\begin{pmatrix}\n 4\\\\\n 5\\\\6\\\\\n\\end{pmatrix}\\beta"



So, the two vectors and any vector will form the linear mapping "T".

Since only the two vectors span the space, the third column is not linearly independent. Any vector that is a linear combination of the other two, will do. So any matrix of the form 

"T=\\begin{pmatrix}\n 1&4&x+4y\\\\\n 2&5&2x+5y\\\\\n 3&6&3x+6y\\\\\n\\end{pmatrix}"


has the designed properties of the linear mapping for any "x, y\\in \\R."


A trivial example is


"T=\\begin{pmatrix}\n 1&4&0\\\\\n 2&5&0\\\\\n 3&6&0\\\\\n\\end{pmatrix}"

Other example


"T=\\begin{pmatrix}\n 1&4&5\\\\\n 2&5&7\\\\\n 3&6&9\\\\\n\\end{pmatrix}"

There are infinitely many possibilities.



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