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 '18 at 18:17
hlineehlinee
755
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add a comment |
$begingroup$
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 '18 at 18:17
hlineehlinee
755
$endgroup$
1
$begingroup$
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.
$endgroup$
– P. Factor
Nov 20 '18 at 18:23
add a comment |
$begingroup$
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 '18 at 18:17
hlineehlinee
755
$endgroup$
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 '18 at 18:17
hlineehlinee
755
asked Nov 20 '18 at 18:17
hlineehlinee
755
edited Nov 20 '18 at 18:22
hlinee
asked Nov 20 '18 at 18:17
hlineehlinee
755
asked Nov 20 '18 at 18:17
hlineehlinee
755
asked Nov 20 '18 at 18:17
hlineehlinee
755
755
1
$begingroup$
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.
$endgroup$
– P. Factor
Nov 20 '18 at 18:23
add a comment |
1
$begingroup$
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.
$endgroup$
– P. Factor
Nov 20 '18 at 18:23
1
1
$begingroup$
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.
$endgroup$
– P. Factor
Nov 20 '18 at 18:23
$begingroup$
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.
$endgroup$
– P. Factor
Nov 20 '18 at 18:23
add a comment |
1 Answer
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$begingroup$
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 '18 at 18:24
Mark McClureMark McClure
23.7k34472
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1 Answer
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1 Answer
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active
oldest
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active
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votes
$begingroup$
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 '18 at 18:24
Mark McClureMark McClure
23.7k34472
$endgroup$
add a comment |
$begingroup$
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 '18 at 18:24
Mark McClureMark McClure
23.7k34472
$endgroup$
add a comment |
$begingroup$
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 '18 at 18:24
Mark McClureMark McClure
23.7k34472
$endgroup$
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 '18 at 18:24
Mark McClureMark McClure
23.7k34472
answered Nov 20 '18 at 18:24
Mark McClureMark McClure
23.7k34472
answered Nov 20 '18 at 18:24
Mark McClureMark McClure
23.7k34472
answered Nov 20 '18 at 18:24
Mark McClureMark McClure
23.7k34472
23.7k34472
add a comment |
add a comment |
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1
$begingroup$
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.
$endgroup$
– P. Factor
Nov 20 '18 at 18:23