review of matrices or a fast introduction. why study matrices? fundamental tools for linear systems
TRANSCRIPT
Review of MatricesOr A Fast Introduction
Why Study Matrices?Fundamental Tools For Linear
Systems
Matrices Come From Systems of Equations
We’ve known how to find
For different right hand sides
Matrices Come From Systems of Equations
Matrices are all about what happens with different
Matrix
Vector
Matrices Come From Systems of Equations
Matrices are all about what happens with different
Matrices Come From Systems of Equations
Fundamental Definition:Matrix Vector
Multiplication
(Think of it as a system of equations!)Try
it!
Matrices Come From Systems of Equations
Fundamental Definition:Matrix Vector
Multiplication
(Think of it as a system of equations!)Try
it!
Matrices Come From Systems of Equations
Fundamental Definition:Matrix Vector
Multiplication
Notation
First IndexIs Row
Second Index is Column
Rows
Columns
Matrices Come From Systems of Equations
Fundamental Definition:Matrix Vector
Multiplication
Notation
Second ColumnFirst
Row
Matrices Come From Systems of Equations
Other Important DefinitionsAdditio
n
Matrices Come From Systems of Equations
Other Important DefinitionsAdditio
n
Matrices Come From Systems of Equations
Other Important DefinitionsAdditio
n
Matrices Come From Systems of Equations
Other Important DefinitionsAdditio
n
Matrices Come From Systems of Equations
Other Important DefinitionsAdditio
n
Matrices Come From Systems of Equations
Other Important DefinitionsAdditio
n
Matrices Come From Systems of Equations
Other Important DefinitionsAdditio
n
In General
Matrices Come From Systems of Equations
Other Important DefinitionsAdditio
n
In General
Matrices Come From Systems of Equations
Other Important DefinitionsAdditio
n
In General
Matrices Come From Systems of Equations
Other Important DefinitionsScalar
Multiplication
Matrices Come From Systems of Equations
Other Important DefinitionsAddition and Scalar Multiplication Give Subtraction
Without a Vector
Matrices Come From Systems of Equations
Other Important DefinitionsMatrix
Multiplication
Matrices Come From Systems of Equations
Other Important DefinitionsMatrix
Multiplication
Matrices Come From Systems of Equations
Other Important DefinitionsMatrix
Multiplication
Matrices Come From Systems of Equations
Other Important DefinitionsMatrix
Multiplication
Matrices Come From Systems of Equations
Other Important DefinitionsMatrix
Multiplication
Matrices Come From Systems of Equations
Other Important DefinitionsMatrix
Multiplication
Matrices Come From Systems of Equations
Other Important DefinitionsMatrix
Multiplication
Special MatricesIdentit
y
Zero
Matrix Algebra RulesA Lot Like Normal Algebra
Rules
Notice:
(Addition Commutes)(Addition Associates)(Multiplication
Associates)(Distribute)
(Scalars Distribute)
(Multiplication
DOES NOTCommute)
Integration and Differentiation
Integration and Differentiation
Matrix InversionMany Matrices Have an
Inverse
Inverse Has Special Properties
For Example:
Inverse FormulaMatrix Inverse For 2
by 2
The Determinant
Inverse Exists If and Only If
Inverse Helps Solve Systems of Equations
Inverse Helps Solve Systems of Equations
Inverse Helps Solve Systems of Equations
Inverse Helps Solve Systems of Equations
Inverse Helps Solve Systems of Equations
Inverse Helps Solve Systems of Equations
Singular MatricesMatrix Inverse For 2
by 2
If the Determinant is 0
The Inverse Does Not Exist!
May Not Have A Solution
Example
Determinant is 0
Impossible!
Really Important Fact
If (and only if) the Determinant is 0
There is a Vector
Where
Example
Three Ways of Saying The Same Thing
(1) The Determinant is 0
(2) The Matrix Has No Inverse (Is Singular)(3) There Is A Vector
Where
Summary
• Matrices Come From Systems Of Equations
• We can do Algebra on Matrices
• Matrices are called “singular” if the determinant is 0. Such matrices have no inverses.
Questions?