# LAWA - Linear Algebra With Applications (2020)

**Lecture notes**.

The notes of the course are updates every week (on Wednesday). Due to the current situation they replace frontal classes.
Please check regularly your email to get updated on the advancement of the course and on the homework due.

## Course Calendar.

**19 ^{th} February**:
Numerical sets, Algebraic structures (groupoids, semigroups, monoids, groups, rings, integral domains).
[From Section 1.1 to Section 1.3.2]

**26 ^{th} February:**
Algebraic structures (fields), Matrices (definitions, sum, product).
[From Section 1.3.3 to Section 2.3]

**4 ^{th} March:**
Matrices (Scalar multiplication, Transposition, Matrix inverse).
[From Section 2.4 to Section 2.6]

**25 ^{th} March:**
Matrices (Diagonal matrices and triangular matrices), Linear equations (Definitions, Systems of linear equations).
[From Section 2.6 to Section 3.2]

**Homework:**Prove Corollary 2.23 and Propositions 2.26 and 2.28; Show why the system in Example 3.4 has no solution while the one in Example 3.5 has infinitely many solutions. [Solutions]

**1 ^{st} April:**
Equivalent systems of linear equations, Gaussian elimination.
[From Section 3.3 to Section 3.4]

**Homework:**Find the reduced row-echelon matrix equivalent to the matrix in Equation (3.11); Using the Gaussian algorithm find all solutions of the system of linear equations in Example 3.18; Determine, without computations but just using the results from Section 3.4, how many solutions have the two systems of linear equations having as augmented matrices the ones in Example 3.15. [Solutions]

**8 ^{th} April:**
Homogeneous systems of linear equations.
[Section 3.5]

**Homework:**Using the Gaussian algorithm, show how to obtain the reduced row-echelon matrix in Example 3.21; Find the general solution to the system of linear equations in Example 3.25. [Solutions]

**15 ^{th} April:**
Inverses of a matrix (The matrix inverse algorithm, inverses and systems of linear equations, conditions for invertibility).
[From Section 4.1 to Section 4.3]

**Homework:**Find the solution of the system of linear equations in Example 4.2 studying the system

*AB=I*; Use the matrix inverse algorithm to find the inverse of the matrix

*A*in Example 4.6; Prove that if

*A,B*are square matrices with

*B*invertible and

*A*, then

^{3}= B*A*is invertible too (Example 4.14). [Solutions]

**22 ^{nd} April:**
Elementary matrices.
[From Section 4.4 to Section 4.5]

**Homework:**Find the inverses of the three elementary matrices in Example 4.18; Find the invertible matrix

*U*(with its decomposition in elementary matrices) and the matrix

*B*in reduced row-echelon form such that

*B = UA*, where

*A*is the matrix in Example 4.23; Find two invertible matrices

*U,V*such that

*UAV*, with

*A*is the matrix in Example 4.26, can be written as the block matrix with the identity matrix on the top-left corner and zeros everywhere else. [Solutions]

**29 ^{th} April:**
Determinant of a matrix.
[From Section 5.1 to Section 5.3]

**Homework:**Find the determinant of the matrix in Example 5.3; Using Theorem 5.9 find the determinants of the matrix in Example 5.11; Prove points

*(2)*and

*(3)*of Theorem 5.18. [Solutions]

**6 ^{th} May:**
The Product Theorem, Cramer's Rule.
[From Section 5.3 to Section 5.4]

**Homework:**Prove point

*(2)*of Theorem 5.22; Find the determinants of the matrix in Example 5.23; Using Cramer's Rule find the solution to the system of linear equations in Theorem 5.32. [Solutions]

**13 ^{th} May:**
Eigenvalues, eigenvectors, diagonalization.
[From Section 5.5 to Section 5.8]

**Homework:**Prove Theorem 5.33; Compute the matrix

*A*in Example 5.45; Show that if two matrices are similar and the first is diagonalizable, then the second is diagonalizable too (Example 5.48). [Solutions]

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