Download Linear Algebra I: Understanding Vectors, Matrices, and Matrix Operations - Prof. Javier A. and more Exams Mechanical Systems Design in PDF only on Docsity! MECE 3349: Linear Algebra I Javier A. Kypuros, Ph.D. Mechanical Engineering UTPA Advanced Organizer Questions ๏What are the differences between vectors and matrices? ๏What arithmetic operations can you do with matrices? ๏Give 3 examples of engineering problems where you might use vectors or matrices. ๏Square matrix (rows by columns) ๏Main diagonal of square matrix ๏Rectangular Matrix (rows by columns) Matrices Vectors Row Vector Column Vector Advanced Organizer Questions ๏What are the differences between scalar and matrix multiplication? ๏What requirements are necessary for two matrices to be multiplicable? ๏Does it matter in what order you multiply matrices? ๏What matrix manipulations are you familiar with? Describe them. ๏Are there any special types of matrices that you are aware of? Matrix Multiplication Matrix Multiplication Rules Use three simple matrices to prove the above rules. Transposition Prove using two simple matrices. Special Matrices Symmetric Skew-Symmetric Upper and Lower Triangular Matrices Upper Triangular Lower Triangular Linear System, Coefficient Matrix, Augmented Matrix Geometric Interpretation of Unique Solution Unique Solution Infinite Solutions No Solution Unique Solutions Gauss Elimination and Back Substitution Upper Triangular Back Substitution Original System of Equations Augmented Matrix After Gaussian Elimination Row 2+ 2 Row 1 Elementary Row Operations ๏Elementary Row Operations for Matrices ‣ Interchange of two rows ‣ Addition of a constant multiple of one row to another row ‣ Multiplication of a row by a nonzero constant ๏Elementary Operations for Equations ‣ Interchange of two equations ‣ Addition of a constant multiple of one equation to another equation ‣ Multiplication of an equation by a nonzero constant ๏Row Equivalent Systems have the same set of solutions A Circuits Example Top Node Bottom Node Left Loop Right Loop Test Your Mettle: A Statics Problem Advanced Organizer Questions ๏How many equations are necessary to solve a linear problem? ๏How do you know if the equations you have are sufficient to solve for all the unknowns? ๏How can you test whether or not the available equations are sufficient to solve the problem? Linear Independence Matrix Rank Row equivalent systems have the same rank. The rank of a system in row echelon form can be readily determined. In row echelon form, the rank is equal to the number of nonzero rows. Linear Independence Continued Test Your Mettle 7.4:5,6,7,&8 Solutions of Linear Systems: Existence, and Uniqueness If solution exists, it can be obtained through Gauss elimination.
Solving Circuit Problem using
Cramer’s Rule
202 1092
iy in +33 =0
20i; + +10i2 = 80
10i2 +25i3 =90
1-1 1
D=|20 10 0
0 10 25
0 -1 1 101 1 -1 0
80 10 0 20 80 0 20 10 80
, 190 10 25] , Jo 90 25] Jo 10 9%
y= D >h= D » and 13 = D
y
Trg
Test Your Mettle 30°
, x
ase / |E
j Gs
iy Fe ¥20 Ib
D E
38.6 Ib
Test Your Mettle General Properties of Determinants 1. Interchange of two rows multiplies the value of the determinant by -1 2. Addition of a multiple of a row to another row does not alter the value of the determinant 3. Multiplication of a row by a nonzero constant multiplies the value of the determinant by that constant 4. (1)-(3) also hold true for columns 5. Transposition does not alter the value of the determinant 6. A zero row or column renders the value of a determinant zero 7. Proportional rows or columns render the value of a determinant zero 8. A matrix is of full rank if its determinant is not zero
Cramer’s Rule
if detA £0
1X1 +412X2 + +++ + Ajnx3 = by
1X1 + a29x2 + +++ + A2nx3 = bo
Gy X1 + AnaX2 + +++ + Annx3 = Dn
dD, D2 Dn
n= D> R= D> vee =
Matrix Inverse
AA!=AlA=I
The inverse of a matrix, A~!, exists if rank A = 7.
The inverse of can be used to solve
a linear system of equations,
Ax=b=>A7!Ax=A'b>x=A~'b.
