# 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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## 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 including two videos at
Find Eigenvectors and Eigenvalues of a 3 by 3 Matrix on Video and Find Eigenvectors and Eigenvalues of a 2 by 2 Matrix on 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 .

## 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 ℝ^{n}.

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.

## 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 .

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.