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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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 yesterday
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
yesterday
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 yesterday
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 yesterday
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 yesterday
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 yesterday
asked yesterday
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 yesterday
hlinee
313
asked yesterday
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.
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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
yesterday
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
yesterday
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
yesterday
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
yesterday
add a comment |
1 Answer
1
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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 yesterday
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 yesterday
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 yesterday
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 yesterday
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 yesterday
Mark McClure
23.1k34170
answered yesterday
Mark McClure
23.1k34170
answered yesterday
Mark McClure
23.1k34170
answered yesterday
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
yesterday