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UnitVector: Unit basis vector along an axis in any dimension—Wolfram Documentation

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UnitVector[k]

gives the two-dimensional unit vector in the k^(th) direction.

UnitVector[n,k]

gives the n-dimensional unit vector in the k^(th) direction.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Applications  
Properties & Relations  
Possible Issues  
See Also
Related Guides
History
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UnitVector

UnitVector[k]

gives the two-dimensional unit vector in the k^(th) direction.

UnitVector[n,k]

gives the n-dimensional unit vector in the k^(th) direction.

Details and Options

  • UnitVector[n,k] is a list of length n with a 1 in position k and 0s elsewhere.
  • UnitVector by default creates a vector containing exact integers.
  • The option WorkingPrecision can be used to specify the precision of vector components.

Examples

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Basic Examples  (2)

The unit vector in the direction in two dimensions:

Wolfram Language code: UnitVector[1]

The unit vector in the direction in three dimensions:

Wolfram Language code: UnitVector[3, 2]

Scope  (2)

A unit vector in dimension 100:

Wolfram Language code: UnitVector[100, 4]

A machine‐precision unit vector in the direction in two dimensions:

Wolfram Language code: N[UnitVector[2]]

A 50‐digit-precision unit vector in the direction in two dimensions:

Wolfram Language code: N[UnitVector[2], 50]

Applications  (3)

Find the matrix for a "black box" linear operator:

Wolfram Language code: m = Table[FourierDST[UnitVector[4, i], "IV"], {i, 4}]

The matrix is equivalent to (though perhaps less efficient than) the "black box":

Wolfram Language code: r = RandomReal[1, 4];
Wolfram Language code: FourierDST[r, "IV"]
Wolfram Language code: m.r

The matrix form allows you to use typical linear algebra functions:

Wolfram Language code: Eigenvalues[m]

p is a random permutation:

Wolfram Language code: p = RandomSample[Range[5]]

Get the permutation matrix:

Wolfram Language code: mp = Map[UnitVector[5, #]&, p]
Wolfram Language code: mp.Range[5]

Compute the unit matrices:

Wolfram Language code: UnitMatrix[{i_, j_}, {m_, n_}] := KroneckerProduct[UnitVector[m, i], UnitVector[n, j]]
Wolfram Language code: UnitMatrix[{2, 3}, {3, 3}]//MatrixForm

Properties & Relations  (1)

A random unit vector:

Wolfram Language code: n = RandomInteger[{1, 1000}]; k = RandomInteger[{1, n}]; v = UnitVector[n, k];

The length is equal to n:

Wolfram Language code: Length[v] == n

There is a 1 in position k:

Wolfram Language code: Position[v, 1] == {{k}}

All other components are zero:

Wolfram Language code: n - Count[v, 0]

Possible Issues  (1)

For very large dimensions n, the vector given by UnitVector may use a lot of memory:

Wolfram Language code: v = UnitVector[1000000, 21]; ByteCount[v]

An alternative is to use a SparseArray to represent the same thing:

Wolfram Language code: s = SparseArray[{21} -> 1, 1000000]
Wolfram Language code: ByteCount[s]
Wolfram Research (2007), UnitVector, Wolfram Language function, https://reference.wolfram.com/language/ref/UnitVector.html (updated 2008).

Text

Wolfram Research (2007), UnitVector, Wolfram Language function, https://reference.wolfram.com/language/ref/UnitVector.html (updated 2008).

CMS

Wolfram Language. 2007. "UnitVector." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2008. https://reference.wolfram.com/language/ref/UnitVector.html.

APA

Wolfram Language. (2007). UnitVector. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/UnitVector.html

BibTeX

@misc{reference.wolfram_2026_unitvector, author="Wolfram Research", title="{UnitVector}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/UnitVector.html}", note=[Accessed: 14-November-2026]}

BibLaTeX

@online{reference.wolfram_2026_unitvector, organization={Wolfram Research}, title={UnitVector}, year={2008}, url={https://reference.wolfram.com/language/ref/UnitVector.html}, note=[Accessed: 14-November-2026]}