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Examples for

Step-by-Step Linear Algebra

Matrix Properties

Compute properties of a matrix step by step:

{{1, -1}, {2, 3}}
{{2,4,4},{1,2,2},{0,7,3}}

Determinants

Find the determinant step by step with various methods:

determinant of {{1,2}, {-1, 2}}
determinant of {{1,2,1}, {1,1,0}, {0,1,1}}

Null Space

Find the null space of a matrix:

null space {{1, 3, 3}, {-3, -5, -3}, {3, 3, 0}}
kernel {{0, 1, 0},{-1, 0, 2},{0, -1, 0}, {0, 0, -1}}

Vector Arithmetic

See how to compute a dot product:

(4, 1) . (-2, 3)

Compute a cross product step by step:

(1,2,3)x(3,4,5)

Linear Independence

Determine the linear independence of vectors step by step:

linear independence {1,0,0},{2,0,0},{0,4,5}
linear independence of (2,x,6),(5y,1,0),(0,0,1)

Matrix Arithmetic

See the steps to multiply matrices:

{{1, 2}, {3, 4}} . {{-1, 1}, {0, 2}}

Invert a matrix one step at a time:

inverse {{1, 1, 2}, {-1, 2, 2}, {3, 2, 3}}

Row Reduction

Write a matrix in reduced row echelon form one step at a time:

{{1,1,5},{1,-1,1}} row reduce
reduced row echelon form: {{1, -3, 3, -4}, {2, 3, -1, 15}, {4, -3, -1, 19}}

Eigenvalues & Eigenvectors

Compute eigenvalues and eigenvectors step by step:

eigenvalues {{3,-1},{0,2}}
eigenvectors {{7,0,-3},{-9,-2,3},{18,0,-8}}

Angle between Vectors

Show the steps to find the angle between two vectors:

angle between {-1,3} and {4,9}
{-1, -3, -7} angle {4, 5, 6}

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Free Unlimited Linear Algebra Practice Problems

RELATED EXAMPLES

  • Linear Algebra
  • Matrices
  • Vectors

    • Rank & Nullity

    Find the rank of a matrix:

    rank {{1, 2, 1}, {-2, -3, 1}, {3, 5, 0}}

    Find the nullity of a matrix:

    nullity {{1, 2}, {3, 4}, {5, 6}, {7, 8}}

    Characteristic Polynomial

    Find the characteristic polynomial of a matrix:

    characteristic polynomial {{1, 2}, {-1, 4}}
    char poly {{10,-35,50,-24},{1,0,0,0},{0,1,0,0},{0,0,1,0}}

    Vector Norms

    Calculate a vector norm or length:

    vector length {2,3,4}
    norm of (y, Sqrt(3), 3x)
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