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
asked 11 hours ago
hlinee
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up vote
6
down vote
favorite
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
asked 11 hours ago
hlinee
313
New contributor
hlinee is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
1
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.
– P. Factor
11 hours ago
add a comment |
up vote
6
down vote
favorite
up vote
6
down vote
favorite
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
asked 11 hours ago
hlinee
313
New contributor
hlinee is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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
asked 11 hours ago
hlinee
313
New contributor
hlinee is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 11 hours ago
hlinee
313
New contributor
hlinee is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 11 hours ago
asked 11 hours ago
hlinee
313
New contributor
hlinee is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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asked 11 hours ago
hlinee
313
asked 11 hours ago
hlinee
313
313
New contributor
hlinee is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
hlinee is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
hlinee is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
1
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.
– P. Factor
11 hours ago
add a comment |
1
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.
– P. Factor
11 hours ago
1
1
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.
– P. Factor
11 hours ago
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.
– P. Factor
11 hours ago
add a comment |
1 Answer
1
active
oldest
votes
up vote
6
down vote
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.
answered 11 hours ago
Mark McClure
23.1k34170
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
6
down vote
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.
answered 11 hours ago
Mark McClure
23.1k34170
add a comment |
up vote
6
down vote
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.
answered 11 hours ago
Mark McClure
23.1k34170
add a comment |
up vote
6
down vote
up vote
6
down vote
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.
answered 11 hours ago
Mark McClure
23.1k34170
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.
answered 11 hours ago
Mark McClure
23.1k34170
answered 11 hours ago
Mark McClure
23.1k34170
answered 11 hours ago
Mark McClure
23.1k34170
answered 11 hours ago
Mark McClure
23.1k34170
23.1k34170
add a comment |
add a comment |
hlinee is a new contributor. Be nice, and check out our Code of Conduct.
hlinee is a new contributor. Be nice, and check out our Code of Conduct.
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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.
– P. Factor
11 hours ago