How is this M(2,2)->R closed under addition?
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
asked Dec 17 at 23:39
isuckatprogramming
256
add a comment |
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
asked Dec 17 at 23:39
isuckatprogramming
256
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
Dec 17 at 23:50
ohhh okay thank you that clarified that for me
– isuckatprogramming
Dec 18 at 1:15
add a comment |
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
asked Dec 17 at 23:39
isuckatprogramming
256
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
asked Dec 17 at 23:39
isuckatprogramming
256
asked Dec 17 at 23:39
isuckatprogramming
256
asked Dec 17 at 23:39
isuckatprogramming
256
asked Dec 17 at 23:39
isuckatprogramming
256
asked Dec 17 at 23:39
isuckatprogramming
256
256
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
Dec 17 at 23:50
ohhh okay thank you that clarified that for me
– isuckatprogramming
Dec 18 at 1:15
add a comment |
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
Dec 17 at 23:50
ohhh okay thank you that clarified that for me
– isuckatprogramming
Dec 18 at 1:15
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
Dec 17 at 23:50
"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
Dec 17 at 23:50
ohhh okay thank you that clarified that for me
– isuckatprogramming
Dec 18 at 1:15
ohhh okay thank you that clarified that for me
– isuckatprogramming
Dec 18 at 1:15
add a comment |
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.
answered Dec 17 at 23:57
Theo Bendit
16.3k12148
add a comment |
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.
answered Dec 17 at 23:44
Lucas Henrique
725313
add a comment |
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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.
answered Dec 17 at 23:57
Theo Bendit
16.3k12148
add a comment |
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.
answered Dec 17 at 23:57
Theo Bendit
16.3k12148
add a comment |
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.
answered Dec 17 at 23:57
Theo Bendit
16.3k12148
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.
answered Dec 17 at 23:57
Theo Bendit
16.3k12148
answered Dec 17 at 23:57
Theo Bendit
16.3k12148
answered Dec 17 at 23:57
Theo Bendit
16.3k12148
answered Dec 17 at 23:57
Theo Bendit
16.3k12148
16.3k12148
add a comment |
add a comment |
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.
answered Dec 17 at 23:44
Lucas Henrique
725313
add a comment |
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.
answered Dec 17 at 23:44
Lucas Henrique
725313
add a comment |
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.
answered Dec 17 at 23:44
Lucas Henrique
725313
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.
answered Dec 17 at 23:44
Lucas Henrique
725313
answered Dec 17 at 23:44
Lucas Henrique
725313
answered Dec 17 at 23:44
Lucas Henrique
725313
answered Dec 17 at 23:44
Lucas Henrique
725313
725313
add a comment |
add a comment |
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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
Dec 17 at 23:50
ohhh okay thank you that clarified that for me
– isuckatprogramming
Dec 18 at 1:15