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 Nov 20 at 18:17
hlinee
634
add a comment |
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 Nov 20 at 18:17
hlinee
634
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
Nov 20 at 18:23
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 Nov 20 at 18:17
hlinee
634
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 Nov 20 at 18:17
hlinee
634
asked Nov 20 at 18:17
hlinee
634
edited Nov 20 at 18:22
asked Nov 20 at 18:17
hlinee
634
asked Nov 20 at 18:17
hlinee
634
asked Nov 20 at 18:17
hlinee
634
634
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
Nov 20 at 18:23
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
Nov 20 at 18:23
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
Nov 20 at 18:23
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
Nov 20 at 18:23
add a comment |
1 Answer
1
active
oldest
votes
up vote
6
down vote
accepted
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 Nov 20 at 18:24
Mark McClure
23.2k34170
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
accepted
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 Nov 20 at 18:24
Mark McClure
23.2k34170
add a comment |
up vote
6
down vote
accepted
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 Nov 20 at 18:24
Mark McClure
23.2k34170
add a comment |
up vote
6
down vote
accepted
up vote
6
down vote
accepted
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 Nov 20 at 18:24
Mark McClure
23.2k34170
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 Nov 20 at 18:24
Mark McClure
23.2k34170
answered Nov 20 at 18:24
Mark McClure
23.2k34170
answered Nov 20 at 18:24
Mark McClure
23.2k34170
answered Nov 20 at 18:24
Mark McClure
23.2k34170
23.2k34170
add a comment |
add a comment |
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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
Nov 20 at 18:23