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
asked 5 hours ago
isuckatprogramming
256
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up vote
2
down vote
favorite
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 5 hours ago
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
5 hours ago
ohhh okay thank you that clarified that for me
– isuckatprogramming
3 hours ago
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
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 5 hours ago
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 5 hours ago
isuckatprogramming
256
asked 5 hours ago
isuckatprogramming
256
asked 5 hours ago
isuckatprogramming
256
asked 5 hours ago
isuckatprogramming
256
asked 5 hours ago
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
5 hours ago
ohhh okay thank you that clarified that for me
– isuckatprogramming
3 hours ago
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
5 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
5 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
5 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
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 5 hours ago
Theo Bendit
16.2k12148
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up vote
0
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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.
answered 5 hours ago
Lucas Henrique
723312
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
6
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.
answered 5 hours ago
Theo Bendit
16.2k12148
add a comment |
up vote
6
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.
answered 5 hours ago
Theo Bendit
16.2k12148
add a comment |
up vote
6
down vote
accepted
up vote
6
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.
answered 5 hours ago
Theo Bendit
16.2k12148
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 5 hours ago
Theo Bendit
16.2k12148
answered 5 hours ago
Theo Bendit
16.2k12148
answered 5 hours ago
Theo Bendit
16.2k12148
answered 5 hours ago
Theo Bendit
16.2k12148
16.2k12148
add a comment |
add a comment |
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.
answered 5 hours ago
Lucas Henrique
723312
add a comment |
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.
answered 5 hours ago
Lucas Henrique
723312
add a comment |
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.
answered 5 hours ago
Lucas Henrique
723312
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 5 hours ago
Lucas Henrique
723312
answered 5 hours ago
Lucas Henrique
723312
answered 5 hours ago
Lucas Henrique
723312
answered 5 hours ago
Lucas Henrique
723312
723312
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
5 hours ago
ohhh okay thank you that clarified that for me
– isuckatprogramming
3 hours ago