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.

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 .
Find Matrix Inverse Using Row Operations
Add, Subtract and Scalar Multiply Matrices .
Multiplication and Power of Matrices
Eigenvalues and Eigenvectors Questions with Solutions
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 .

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 and 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 .

Linear Algebra Calculators

Find the Inverse of a Matrix Using Row Reduction .
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 .