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What is the syntax to add a vector v1 to each vector in a list of vectors v2?

I know it has to be simple, but I really have searched and not found it.

v1 = {a, b, c}
v2 = {{d, e, f}, {g, h, i}, {j, k, l}}

i.e., sum them in a way to give:

{{a + d, b + e,c + f}, {a + g, b + h, c + i}, {a + j, b + k, c + l}}
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10 Answers 10

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I recommend using Transpose twice since it is more efficient than other approaches. Moreover Plus has the Listable attribute, thus one need not map Plus over a list (vector).

Transpose[v1 + Transpose[v2]]

{{a + d, b + e, c + f}, {a + g, b + h, c + i}, {a + j, b + k, c + l}}

Having said that remember that one can rewrite it very concisely in the Front-End: Esc tr Esc :

enter image description here

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  • $\begingroup$ Hah, clearly my knowledge of Inner isn't well enough entrenched yet. I failed to notice that my Inner effectively simplified to Transpose @* Plus! $\endgroup$
    – Patrick Stevens
    Sep 19, 2015 at 22:28
  • $\begingroup$ @ciao Thanks, unfortunately someone unupvoted my answer (it was edited) so I guess it is inadvisable to improve posts. $\endgroup$
    – Artes
    Sep 20, 2015 at 0:23
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    $\begingroup$ @Artes Of course it is advisable to improve posts! $\endgroup$
    – Dr. belisarius
    Sep 20, 2015 at 18:24
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    $\begingroup$ In similar contexts ... While I use Transpose all the time - for intuitive clarity and readability, sometimes one can get faster performance using Part and All in its place ... just FYI testData = Table[{{i, 2 i}, 3 i}, {i, 1000000}]; Print@Timing[ Length[ Last[ Transpose[testData]]]]; Print@Timing[ Length[ Part[ testData, All, 2]]]; Print@Timing[ Length[ testData[[ All, 2]]]]; $\endgroup$
    – Mark Samuel Tuttle
    Sep 23, 2015 at 13:32
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To achieve what you need requires to distribute the sum over v2:

(v1 + # &) /@ v2

which is a short form of:

Map[ v1 + # &, v2 ]
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  • $\begingroup$ This is a good one for my list of "expressions that help me understand mathematica syntax"! Thanks. $\endgroup$
    – DrBubbles
    Sep 19, 2015 at 22:17
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Alternative method using the magic that is Inner:

Inner[Plus, {a, b, c}, Transpose@{{d, e, f}, {g, h, i}, {j, k, l}}, List]
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Also:

Plus @@@ Thread[{v1, v2}, List, {2}]
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I couldn't resist adding this:

TranslationTransform[v1] /@ v2
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    $\begingroup$ More terse and universal version: TranslationTransform[v1][v2]. $\endgroup$
    – Alexey Popkov
    Oct 12, 2015 at 13:39
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    $\begingroup$ This function is for that purpose by far the slowest solution ... $\endgroup$
    – mrz
    May 27, 2016 at 21:10
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Using Outer is also an efficient approach, in some cases faster than using Transpose twice, the undocumented function Statistics`Library`MatrixRowTranslate may be the fastest

v1 = RandomReal[1, 10^5];
v2 = RandomReal[1, {10^3, 10^5}];

r1 = Transpose[v1 + Transpose[v2]]; // RepeatedTiming
r2 = Outer[Plus, {v1}, v2, 1][[1]]; // RepeatedTiming
r3 = Block[{res=v2}, Statistics`Library`MatrixRowTranslate[res, v1]; res]; // RepeatedTiming

r1===r2===r3

Output

{0.541983, Null}
{0.446791, Null}
{0.241258, Null}
True

Related link:
Improving Map Function on Lists

Elegant operations on matrix rows and columns

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In versions 13.1+ you can wrap v1 with Threaded:

Threaded[v1] + v2

{{a + d, b + e, c + f}, {a + g, b + h, c + i}, {a + j, b + k, c + l}}
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  • $\begingroup$ Interesting new command! $\endgroup$
    – bill s
    Apr 11 at 22:27
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Total[Tuples[{{v1}, v2}], {2}]

 {{a + d, b + e, c + f}, {a + g, b + h, c + i}, {a + j, b + k, c + l}}

Distribute[{{v1}, v2}, List, List, List, Plus]

 {{a + d, b + e, c + f}, {a + g, b + h, c + i}, {a + j, b + k, c + l}}
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In versions 13.1+ you can use ReplaceAt:

ReplaceAt[{v2}, _ -> Map[#1 + v1 &], 0]

(*{{a + d, b + e, c + f}, {a + g, b + h, c + i}, {a + j, b + k, c + l}}*)
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Here was my first thought...

ConstantArray[v1, 3] + v2

or more generally

ConstantArray[v1, Length[v2]] + v2
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