Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.
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Transcript of Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.
![Page 1: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/1.jpg)
Review of MatricesOr A Fast Introduction
![Page 2: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/2.jpg)
Why Study Matrices?Fundamental Tools For Linear
Systems
![Page 3: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/3.jpg)
Matrices Come From Systems of Equations
We’ve known how to find
For different right hand sides
![Page 4: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/4.jpg)
Matrices Come From Systems of Equations
Matrices are all about what happens with different
Matrix
Vector
![Page 5: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/5.jpg)
Matrices Come From Systems of Equations
Matrices are all about what happens with different
![Page 6: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/6.jpg)
Matrices Come From Systems of Equations
Fundamental Definition:Matrix Vector
Multiplication
(Think of it as a system of equations!)Try
it!
![Page 7: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/7.jpg)
Matrices Come From Systems of Equations
Fundamental Definition:Matrix Vector
Multiplication
(Think of it as a system of equations!)Try
it!
![Page 8: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/8.jpg)
Matrices Come From Systems of Equations
Fundamental Definition:Matrix Vector
Multiplication
Notation
First IndexIs Row
Second Index is Column
Rows
Columns
![Page 9: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/9.jpg)
Matrices Come From Systems of Equations
Fundamental Definition:Matrix Vector
Multiplication
Notation
Second ColumnFirst
Row
![Page 10: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/10.jpg)
Matrices Come From Systems of Equations
Other Important DefinitionsAdditio
n
![Page 11: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/11.jpg)
Matrices Come From Systems of Equations
Other Important DefinitionsAdditio
n
![Page 12: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/12.jpg)
Matrices Come From Systems of Equations
Other Important DefinitionsAdditio
n
![Page 13: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/13.jpg)
Matrices Come From Systems of Equations
Other Important DefinitionsAdditio
n
![Page 14: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/14.jpg)
Matrices Come From Systems of Equations
Other Important DefinitionsAdditio
n
![Page 15: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/15.jpg)
Matrices Come From Systems of Equations
Other Important DefinitionsAdditio
n
![Page 16: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/16.jpg)
Matrices Come From Systems of Equations
Other Important DefinitionsAdditio
n
In General
![Page 17: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/17.jpg)
Matrices Come From Systems of Equations
Other Important DefinitionsAdditio
n
In General
![Page 18: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/18.jpg)
Matrices Come From Systems of Equations
Other Important DefinitionsAdditio
n
In General
![Page 19: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/19.jpg)
Matrices Come From Systems of Equations
Other Important DefinitionsScalar
Multiplication
![Page 20: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/20.jpg)
Matrices Come From Systems of Equations
Other Important DefinitionsAddition and Scalar Multiplication Give Subtraction
Without a Vector
![Page 21: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/21.jpg)
Matrices Come From Systems of Equations
Other Important DefinitionsMatrix
Multiplication
![Page 22: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/22.jpg)
Matrices Come From Systems of Equations
Other Important DefinitionsMatrix
Multiplication
![Page 23: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/23.jpg)
Matrices Come From Systems of Equations
Other Important DefinitionsMatrix
Multiplication
![Page 24: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/24.jpg)
Matrices Come From Systems of Equations
Other Important DefinitionsMatrix
Multiplication
![Page 25: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/25.jpg)
Matrices Come From Systems of Equations
Other Important DefinitionsMatrix
Multiplication
![Page 26: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/26.jpg)
Matrices Come From Systems of Equations
Other Important DefinitionsMatrix
Multiplication
![Page 27: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/27.jpg)
Matrices Come From Systems of Equations
Other Important DefinitionsMatrix
Multiplication
![Page 28: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/28.jpg)
Special MatricesIdentit
y
Zero
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Matrix Algebra RulesA Lot Like Normal Algebra
Rules
Notice:
(Addition Commutes)(Addition Associates)(Multiplication
Associates)(Distribute)
(Scalars Distribute)
(Multiplication
DOES NOTCommute)
![Page 30: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/30.jpg)
Integration and Differentiation
![Page 31: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/31.jpg)
Integration and Differentiation
![Page 32: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/32.jpg)
Matrix InversionMany Matrices Have an
Inverse
Inverse Has Special Properties
For Example:
![Page 33: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/33.jpg)
Inverse FormulaMatrix Inverse For 2
by 2
The Determinant
Inverse Exists If and Only If
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Inverse Helps Solve Systems of Equations
![Page 35: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/35.jpg)
Inverse Helps Solve Systems of Equations
![Page 36: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/36.jpg)
Inverse Helps Solve Systems of Equations
![Page 37: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/37.jpg)
Inverse Helps Solve Systems of Equations
![Page 38: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/38.jpg)
Inverse Helps Solve Systems of Equations
![Page 39: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/39.jpg)
Inverse Helps Solve Systems of Equations
![Page 40: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/40.jpg)
Singular MatricesMatrix Inverse For 2
by 2
If the Determinant is 0
The Inverse Does Not Exist!
May Not Have A Solution
![Page 41: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/41.jpg)
Example
Determinant is 0
Impossible!
![Page 42: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/42.jpg)
Really Important Fact
If (and only if) the Determinant is 0
There is a Vector
Where
![Page 43: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/43.jpg)
Example
![Page 44: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/44.jpg)
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
![Page 45: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/45.jpg)
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.
![Page 46: Review of Matrices Or A Fast Introduction. Why Study Matrices? Fundamental Tools For Linear Systems.](https://reader033.fdocuments.us/reader033/viewer/2022051401/56649e8f5503460f94b93b89/html5/thumbnails/46.jpg)
Questions?