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Linear Algebra

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The algebra and geometry of finite dimensional vector spaces; determinants; linear transformations and matrices; characteristic values and vectors of linear transformations.











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Uploaded on
November 22, 2025
Number of pages
21
Written in
2022/2023
Type
Class notes
Professor(s)
Dr. abernathy
Contains
All classes

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Due Thurs (1119)
- Orientation assignment

- HW 1.1

(X, X,...a) Linear Equation: one that can be
written in the form:
Citi & Ce Ket... + Can = b

coefficients (real#s)

Ex: 2x, + 3×2 = 7
- ✗it ½xz-✗ 3 = 11



System of Equations: Multiple linear

equations in same variables


Ex: Six,-✗2=5
3×1 + X2 = I


2x.-✗ 2=5
+
3x, + ✗ 2 = / 3(5) txz=l
1 8 ×2=1151
5-1
Sx, = 6
5 S -¥ -⅓
I 1=65
✗ 2 =-Bg

,Consistent: System w/@least I solution
- I solution, many solutions, etc. Q: Is EF

inconsistent: system w/no solution [2416
- 2 30
Q: Does a solution exist?
② Is that solution unique? (is it the only one?) NOTE: Any
Matrix is now
MX h equivalent t

row column input many E
matrices.
Echelon Form A rectangular matrix is in

echelon form if:
Reduct
① All non-Zero rows are above. any Zero rows.
How to
② Each leading non-Zero entry of a row is in

, Vector: ordered list of #s Linear Combination

(5,0) (3,211) (¾ Have vectors ,Ja,-

Then:
Two Vectors not equal: C.lit Cziet..-tepi

(Scalars (weights) C
(8) + (s)

Vectors in 15h"

R"-1:?)/ aier to all isia.nl Ex.

(all poss. Vectors w/n entries) l Choose any sca

Adding vectors: Scale Vectors:
linear comb. of
* Must be same size* ☒¾ Julti. each entry by scalar

Z 13
-
7
-
:)
( ☐ f) + ☐ E.
Scalar
(must be
add component by component a real #)
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