How is this M(2,2)->R closed under addition?











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so I understand that It is closed under scalar multiplication, but how is it closed under addition? what if I add another matrix to A? and create new a,b,c,d values?






linear-algebra







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    "Closed under addition" is not quite the right phrase here. We use the properties of being closed under scalar multiplication and closed under addition to define vector spaces (with other properties), but the issue here is whether $T$ is a linear transformation, i.e. a mapping from one vector space to another that preserves operations of scalar multiplication and vector addition.
    – hardmath
    4 hours ago










  • ohhh okay thank you that clarified that for me
    – isuckatprogramming
    3 hours ago















up vote
1
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enter image description here



so I understand that It is closed under scalar multiplication, but how is it closed under addition? what if I add another matrix to A? and create new a,b,c,d values?






linear-algebra







share|cite|improve this question


















  • 1




    "Closed under addition" is not quite the right phrase here. We use the properties of being closed under scalar multiplication and closed under addition to define vector spaces (with other properties), but the issue here is whether $T$ is a linear transformation, i.e. a mapping from one vector space to another that preserves operations of scalar multiplication and vector addition.
    – hardmath
    4 hours ago










  • ohhh okay thank you that clarified that for me
    – isuckatprogramming
    3 hours ago













up vote
1
down vote

favorite









up vote
1
down vote

favorite











enter image description here



so I understand that It is closed under scalar multiplication, but how is it closed under addition? what if I add another matrix to A? and create new a,b,c,d values?






linear-algebra







share|cite|improve this question













enter image description here



so I understand that It is closed under scalar multiplication, but how is it closed under addition? what if I add another matrix to A? and create new a,b,c,d values?






linear-algebra




linear-algebra






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asked 4 hours ago









isuckatprogramming

206




206








  • 1




    "Closed under addition" is not quite the right phrase here. We use the properties of being closed under scalar multiplication and closed under addition to define vector spaces (with other properties), but the issue here is whether $T$ is a linear transformation, i.e. a mapping from one vector space to another that preserves operations of scalar multiplication and vector addition.
    – hardmath
    4 hours ago










  • ohhh okay thank you that clarified that for me
    – isuckatprogramming
    3 hours ago














  • 1




    "Closed under addition" is not quite the right phrase here. We use the properties of being closed under scalar multiplication and closed under addition to define vector spaces (with other properties), but the issue here is whether $T$ is a linear transformation, i.e. a mapping from one vector space to another that preserves operations of scalar multiplication and vector addition.
    – hardmath
    4 hours ago










  • ohhh okay thank you that clarified that for me
    – isuckatprogramming
    3 hours ago








1




1




"Closed under addition" is not quite the right phrase here. We use the properties of being closed under scalar multiplication and closed under addition to define vector spaces (with other properties), but the issue here is whether $T$ is a linear transformation, i.e. a mapping from one vector space to another that preserves operations of scalar multiplication and vector addition.
– hardmath
4 hours ago




"Closed under addition" is not quite the right phrase here. We use the properties of being closed under scalar multiplication and closed under addition to define vector spaces (with other properties), but the issue here is whether $T$ is a linear transformation, i.e. a mapping from one vector space to another that preserves operations of scalar multiplication and vector addition.
– hardmath
4 hours ago












ohhh okay thank you that clarified that for me
– isuckatprogramming
3 hours ago




ohhh okay thank you that clarified that for me
– isuckatprogramming
3 hours ago










2 Answers
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Help me fill this in:



begin{align*}
T begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
T begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
T begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} +T begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} + begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in M_{2, 2} \
Tleft(begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} + begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix}right) &= underline{hspace{50pt}} in mathbb{R}.
end{align*}
Are your answers to questions 3 and 5 the same? If so, then $T$ is additive. You can try a similar tactic with scalar multiplication. Let me know with a comment if you want further help.






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    TIP: the sum of two matrices is a matrix. Try writing $T(A)$ in terms of the values of this matrix and, after expanding each term (because they're bound on the other two matrices), compare the final expression two the other two expressions.






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      2 Answers
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      2 Answers
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      Help me fill this in:



      begin{align*}
      T begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
      T begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
      T begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} +T begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
      begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} + begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in M_{2, 2} \
      Tleft(begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} + begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix}right) &= underline{hspace{50pt}} in mathbb{R}.
      end{align*}
      Are your answers to questions 3 and 5 the same? If so, then $T$ is additive. You can try a similar tactic with scalar multiplication. Let me know with a comment if you want further help.






      share|cite|improve this answer

























        up vote
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        Help me fill this in:



        begin{align*}
        T begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
        T begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
        T begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} +T begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
        begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} + begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in M_{2, 2} \
        Tleft(begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} + begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix}right) &= underline{hspace{50pt}} in mathbb{R}.
        end{align*}
        Are your answers to questions 3 and 5 the same? If so, then $T$ is additive. You can try a similar tactic with scalar multiplication. Let me know with a comment if you want further help.






        share|cite|improve this answer























          up vote
          5
          down vote



          accepted







          up vote
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          down vote



          accepted






          Help me fill this in:



          begin{align*}
          T begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
          T begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
          T begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} +T begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
          begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} + begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in M_{2, 2} \
          Tleft(begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} + begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix}right) &= underline{hspace{50pt}} in mathbb{R}.
          end{align*}
          Are your answers to questions 3 and 5 the same? If so, then $T$ is additive. You can try a similar tactic with scalar multiplication. Let me know with a comment if you want further help.






          share|cite|improve this answer












          Help me fill this in:



          begin{align*}
          T begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
          T begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
          T begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} +T begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
          begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} + begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in M_{2, 2} \
          Tleft(begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} + begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix}right) &= underline{hspace{50pt}} in mathbb{R}.
          end{align*}
          Are your answers to questions 3 and 5 the same? If so, then $T$ is additive. You can try a similar tactic with scalar multiplication. Let me know with a comment if you want further help.







          share|cite|improve this answer












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          share|cite|improve this answer










          answered 4 hours ago









          Theo Bendit

          16.2k12148




          16.2k12148






















              up vote
              0
              down vote













              TIP: the sum of two matrices is a matrix. Try writing $T(A)$ in terms of the values of this matrix and, after expanding each term (because they're bound on the other two matrices), compare the final expression two the other two expressions.






              share|cite|improve this answer

























                up vote
                0
                down vote













                TIP: the sum of two matrices is a matrix. Try writing $T(A)$ in terms of the values of this matrix and, after expanding each term (because they're bound on the other two matrices), compare the final expression two the other two expressions.






                share|cite|improve this answer























                  up vote
                  0
                  down vote










                  up vote
                  0
                  down vote









                  TIP: the sum of two matrices is a matrix. Try writing $T(A)$ in terms of the values of this matrix and, after expanding each term (because they're bound on the other two matrices), compare the final expression two the other two expressions.






                  share|cite|improve this answer












                  TIP: the sum of two matrices is a matrix. Try writing $T(A)$ in terms of the values of this matrix and, after expanding each term (because they're bound on the other two matrices), compare the final expression two the other two expressions.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 4 hours ago









                  Lucas Henrique

                  723312




                  723312






























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