What does it mean geometrically to add two matrices?












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If you think of matrix-vector multiplication geometrically as a linear transformation to a new coordinate system and matrix-matrix multiplication as the composition of two separate linear transformations, what does it mean to add two matrices together?



Would it make sense to think of it in terms of adding each basis vector separately to create a new set of basis vectors?






linear-algebra matrices







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    $begingroup$
    You can think of it as adding basis vectors. But beware that after the addition, the new set of vectors may not be independent any more.
    $endgroup$
    – P. Factor
    Nov 20 '18 at 18:23
















6












$begingroup$


If you think of matrix-vector multiplication geometrically as a linear transformation to a new coordinate system and matrix-matrix multiplication as the composition of two separate linear transformations, what does it mean to add two matrices together?



Would it make sense to think of it in terms of adding each basis vector separately to create a new set of basis vectors?






linear-algebra matrices







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    You can think of it as adding basis vectors. But beware that after the addition, the new set of vectors may not be independent any more.
    $endgroup$
    – P. Factor
    Nov 20 '18 at 18:23














6












6








6


1



$begingroup$


If you think of matrix-vector multiplication geometrically as a linear transformation to a new coordinate system and matrix-matrix multiplication as the composition of two separate linear transformations, what does it mean to add two matrices together?



Would it make sense to think of it in terms of adding each basis vector separately to create a new set of basis vectors?






linear-algebra matrices







share|cite|improve this question











$endgroup$




If you think of matrix-vector multiplication geometrically as a linear transformation to a new coordinate system and matrix-matrix multiplication as the composition of two separate linear transformations, what does it mean to add two matrices together?



Would it make sense to think of it in terms of adding each basis vector separately to create a new set of basis vectors?






linear-algebra matrices




linear-algebra matrices






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 20 '18 at 18:22







hlinee

















asked Nov 20 '18 at 18:17









hlineehlinee

755




755








  • 1




    $begingroup$
    You can think of it as adding basis vectors. But beware that after the addition, the new set of vectors may not be independent any more.
    $endgroup$
    – P. Factor
    Nov 20 '18 at 18:23














  • 1




    $begingroup$
    You can think of it as adding basis vectors. But beware that after the addition, the new set of vectors may not be independent any more.
    $endgroup$
    – P. Factor
    Nov 20 '18 at 18:23








1




1




$begingroup$
You can think of it as adding basis vectors. But beware that after the addition, the new set of vectors may not be independent any more.
$endgroup$
– P. Factor
Nov 20 '18 at 18:23




$begingroup$
You can think of it as adding basis vectors. But beware that after the addition, the new set of vectors may not be independent any more.
$endgroup$
– P. Factor
Nov 20 '18 at 18:23










1 Answer
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$begingroup$

Linearity works both ways. That is,
$$
(A+B)vec{v} = Avec{v} + Bvec{v}.
$$
Thus, you can think of the linear transformation defined by $A+B$ as applied to the vector $vec{v}$ as addition of the images under $A$ and $B$, separately, added together.






share|cite|improve this answer









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    1 Answer
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    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    6












    $begingroup$

    Linearity works both ways. That is,
    $$
    (A+B)vec{v} = Avec{v} + Bvec{v}.
    $$
    Thus, you can think of the linear transformation defined by $A+B$ as applied to the vector $vec{v}$ as addition of the images under $A$ and $B$, separately, added together.






    share|cite|improve this answer









    $endgroup$


















      6












      $begingroup$

      Linearity works both ways. That is,
      $$
      (A+B)vec{v} = Avec{v} + Bvec{v}.
      $$
      Thus, you can think of the linear transformation defined by $A+B$ as applied to the vector $vec{v}$ as addition of the images under $A$ and $B$, separately, added together.






      share|cite|improve this answer









      $endgroup$
















        6












        6








        6





        $begingroup$

        Linearity works both ways. That is,
        $$
        (A+B)vec{v} = Avec{v} + Bvec{v}.
        $$
        Thus, you can think of the linear transformation defined by $A+B$ as applied to the vector $vec{v}$ as addition of the images under $A$ and $B$, separately, added together.






        share|cite|improve this answer









        $endgroup$



        Linearity works both ways. That is,
        $$
        (A+B)vec{v} = Avec{v} + Bvec{v}.
        $$
        Thus, you can think of the linear transformation defined by $A+B$ as applied to the vector $vec{v}$ as addition of the images under $A$ and $B$, separately, added together.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 20 '18 at 18:24









        Mark McClureMark McClure

        23.7k34472




        23.7k34472






























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