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.

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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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)
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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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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