Linear Algebra - Questions with Solutions

Linear algebra questions with solutions and detailed explanations on matrices , spaces, subspaces and vectors , determinants , systems of linear equations and online linear algebra calculators are included.
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Interactive Math Tools for Exploring Concepts, Skills and Theorems

Matrices

• Matrices with Examples and Questions with Solutions.
• Transpose of a Matrix.
• Symmetric Matrix.
• Identity Matrix.
• Diagonal Matrices.
• Triangular Matrix.
• Diagonalization of Matrices.
• Inverse Matrix Questions with Solutions.
• >Inverse of Matrix Products
• Find Matrix Inverse Using Row Operations
• Add, Subtract and Scalar Multiply Matrices.
• Multiplication and Power of Matrices
• Eigenvalues and Eigenvectors Questions with Solutions
• Find Eigenvectors and Eigenvalues of a 3 by 3 Matrix on Video (Video)
• Find Eigenvectors and Eigenvalues of a 2 by 2 Matrix on Video (Video)
• Row Operations and Elementary Matrices.
• The Three Row Operations on Augmented Matrices .
• Write a Matrix in Reduced Row Echelon Form.
• Pivots of a Matrix in Row Echelon Form - Examples with Solutions.
• Null Space and Nullity of a Matrix .
• Column and Row Spaces and Rank of a Matrix .
• Free and Basic Variables of a Matrix - Examples with Solutions.
• Orthogonal Matrices - Examples with Solutions.
• The QR Decomposition of a Matrix.
• LU Decomposition of a Matrix.
• Properties of Matrix Operations .

Applications of Matrices

• Matrix Transformations in Computer Graphics.

Least Squares Problems

• Solve Least Squares Problems by the Normal Equations.
• Solve Least Squares Problems by the QR Decomposition.

Spaces, Subspaces and Vectors

• Vector Spaces - Examples with Solutions.
• Subspaces - Examples with Solutions.
• Vectors in ℝⁿ.
• Inner Product, Orthogonality and Length of Vectors .
• Orthogonal Vectors - Examples with Solutions .
• Linear Combinations of Vectors.
• Span of Vectors.
• Linearly Independent and Dependent Vectors - Examples with Solutions .
• Testing for Linearity of Vectors in a Subspace - Examples with Solutions .
• Basis, Coordinates and Dimension of Vector Spaces .
• Change of Basis - Examples with Solutions .
• Orthonormal Basis - Examples with Solutions .
• The Gram Schmidt Process for Orthonormal Basis . Examples with Solutions

determinants

• Determinant of a Square Matrix.
• Find Determinant Using Row Reduction.

Systems of Linear Equations

• The Three Elementary Operations on Systems.
• Gaussian Elimination to Solve Systems - Questions with Solutions.
• The Elimination Method in Systems - Questions with Solutions.
• Cramer's Rule with Questions and Solutions.

Videos on Linear Algebra

• Find Eigevectors and Eigenvalues of a 2 by 2 Matrix .
• Solve a 2 by 2 System of Equations by Elimination .
• Gaussian Elimination to Solve a 3 by 3 System of Equations .
• Inverse of 3 by 3 Matrix Using Gauss-Jordan .

Linear Algebra Calculators

• Find the Inverse of a Matrix Using Row Reduction .
• Multiply Matrix Calculator .
• Add Matrices Calculator .
• Row Echelon Form Calculator .
• Row Reduce Agmented Matrices - Calculator .
• Online Calculator for The Determinant of a Matrix of Any Size .
• QR Decomposition of Matrices Calculator .
• Online LU Decomposition of a Matrix Calculator.
• Systems of Equations with Complex Coefficients Solver.
• Matrix Calculator Multiplication ; step by step calculator for educational purposes.
• Interactive Math Tools for Exploring Concepts and Theorems