Examples for

Linear Algebra

Linear algebra uses the tools and methods of vector and matrix operations to determine the properties of linear systems. Wolfram|Alpha's rigorous computational knowledge of topics such as vectors, vector spaces and matrix theory is a great resource for calculating and exploring the properties of vectors and matrices, the linear independence of vectors and the vector spaces underlying sets of vectors and matrices.
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Vectors

Find properties of and perform computations on n-dimensional vectors.

Compute properties of a vector:

vector {2, -5, 4}

Compute a cross product:

{1/4, -1/2, 1} cross {1/3, 1, -2/3}
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Matrices

Explore various properties of matrices.

Calculate properties of a matrix:

{{6, -7}, {0, 3}}

Multiply matrices:

{{2, -1}, {1, 3}} . {{1, 2}, {3, 4}}

Row reduce a matrix:

row reduce {{2, 1, 0, -3}, {3, -1, 0, 1}, {1, 4, -2, -5}}
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Linear Independence

Check vectors for both linear dependence and linear independence.

Determine whether a set of vectors is linearly independent:

Are (2, -1) and (4, 2) linearly independent?
linear independence (1, 3, -2), (2, 1, -3), (-3, 6, 3)

Specify complex vectors:

are (1, i), (i, -1) linearly independent?

Specify vectors with one or more symbolic components:

linear independence of {(1, 3, -1), (-1, -5, 5), (4, 7, h)}
linear independence (a, b, c, d), (e, f, g, h), (i, j, k, l)
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Vector Spaces

Compute properties of linear vector spaces.

Compute the row space of a matrix:

row space {{1, 2, -5}, {-1, 0, -1}, {2, 1, -1}}

Compute the column space of a matrix:

{{1, 0, -1}, {2, -1, 3}} column space

Compute the null space of a matrix:

null space of {{1, 0, -2, 1}, {2, -1, 1, 0}, {0, 2, -3, 1}}
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