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Gauss-Jordan - variants
ä First: Pivoting can be implemented just like Gaussian elimination.
Important: Never swap a row with a row above it! (de- stroys structure) Always swap a row with a row below it (when interchange is needed).
Common variant: After an elimination step is completed divide the row by diagonal entry akk (→ at the end all diagonal entries are ones) ..
- Redo the previous example with this variant.
- Question: is this more or less costly than original method?
NOTE: unless otherwise specified Gauss-Jordan will be the standard one (no scaling by diagonal). 37
Linear systems – summary of complexity results
ä The number of operations needed to solve a triangular linear system with n unknowns is CT (n) = n^2
ä The number of operations required to solve a linear system with n unknowns by Gaussian elimination is CG(n) ≈ 23 n^3
ä The number of operations required to solve a linear system with n unknowns by Gauss-Jordan elimination is CGJ (n) ≈ n^3
ä Note: remember that Gauss-Jordan costs 50% more than Gauss.
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VECTOR EQUATIONS [PARTS OF 1.3]
Vectors and the set Rn
ä A vector of dimension n is an ordered list of n numbers Example:
v =
;^ w^ =
[
]
; z =
ä v is in R^3 , w is in R^2 and z is in R? ä In R^3 the R stands for the set of real numbers that appear as entries in the vector, and the exponents 3, indicate that each vector contains 3 entries. ä A vector can be viewed just as a matrix of dimension m × 1
C-2 Docsity.com
ä Rn^ is the set of all vectors of dimension n. We will see later that this is a vector space, i.e., a set that has some special properties with respect to operations on vectors.
ä Two vectors in Rn^ are equal if and only if their corre- sponding entries are equal.
[Note: what does if and only if mean? – find out]
ä Given two vectors u and v in Rn^ , their sum is the vector u + v obtained by adding corresponding entries of u and v
ä Given a vector u and a real number α, the scalar multiple of u by α is the vector αu obtained by multiplying each entry in u by α
ä Let us look at this in detail
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Sum of two vectors ä (!) Note: the two vectors must be of the same dimen- sion
x =
x 1 x 2 x 3
;^ y^ =
y 1 y 2 y 3
;^ →^ x^ +^ y^ =
x 1 + y 1 y 2 + x 2 x 3 + y 3
with numbers:
x =
;^ y^ =
;^ →^ x^ +^ y^ =
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Multiplication by a scalar
ä Given: a number α (a ’scalar’) and a vector x:
α ∈ R, x ∈ R^3 , → αx =
αx 1 αx 2 αx 3
with numbers:
α = 4; x =
→^ αx^ =
In the text vectors are represented by bold characters and scalars by light characters. We will often use Greek letters for scalars and regular latin symbols for vectors
Properties of + and α∗
ä The vector whose entries are all zero is called the zero vector and is denoted by 0.
- (a) x + y = y + x (Addition is commutative)
- (b) x + (y + z) = (x + y) + z (Addition is associative)
- (c) 0 + x = x + 0 = x, ( 0 is the vector of all zeros)
- (d) x + (−x) = −x + x = 0 (−x is the vector (−1)x)
- (e) α(x + y) = αx + αy
- (f) (α + β)x = αx + βx
- (g) (αβ)x = α(βx)
- (h) 1 x = x for any x
Geometric interpretation of addition of 2 vectors
First viewpoint:
Think of moving (“rigidly”) one of the vectors so its origin is at endpoint of the other vector. Then x + y is the vector from origin to the end point of the shifted vector.
y x shifted
origin
y with
x
x+y
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Second viewpoint:
x + y correponds to the fourth vertex of the parallelogram whose other three vertices are: O, x, and y
y x y
O
x
x+y
- Using the first viewpoint, show geometrically how to add the vectors[ 1 1
]
[
]
; and
[
]
C-
Linear combinations
ä Very important concept ..
A linear combination of m vectors is a vector of the form: x = α 1 x 1 + α 2 x 2 + · · · + αmxm where α 1 , α 2 , · · · , αm, are scalars and x 1 , x 2 , · · · , xm, are vectors in Rn.
ä The scalars α 1 , α 2 , · · · , αm are called the weights of the linear combination
ä They can be any real numbers, including zero
Linear combinations
Example: Linear combinations of vectors in R^3 :
u = 2
;^ w^ = 2
And we have: u^ =
;^ w^ =
Note: for w the second weight is − 1 and the third is +1.
The linear span of a set of vectors
Definition: If v 1 , · · · , vp are in Rn, then the set of all linear combinations of v 1 , · · · , vp is denoted by span{v 1 , · · · , vp} and is called the subset of Rn^ spanned (or generated) by v 1 , · · · , vp. That is, span{v 1 , · · · , vp} is the collection of all vectors that can be written in the form α 1 v 1 + α 2 v 2 + · · · + αpvp with α 1 , α 2 , · · · , αp scalars.
- What is span{u} in R^2 where u =
[
]
- What is span{v} in R^2 where v =
[
]
- What is span{u, v} in R^2 with u, v given above?
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[
]
belong to this span{u, v}?
- Same question for the vector
[
]
- What is span{u, v} in R^3 when:
u =
;^ v^ =
a =
;^ b^ =
belong to span{u, v} found in previous quest.?
- Is span{u, v} the same as span{v, u}?
- Is span{u, v} the same as span{ 2 u, − 3 v}? C-
Geometric interpretation of span{v}
ä Let v be a nonzero vector inR^3
ä Then span{v} is the set of all scalar multiples of v
ä This is also the set of points on the line in R^3 through v and 0.
(Figure 1.0 from text).
Geometric interpretation of span{u, v}
ä Let u, v be two nonzero vectors in R^3 with v not a multiple of u.
ä Then span{u, v} is the plane in R^3 that contains u, v, and 0. ä In particular, span{u, v} contains the two lines span{u} and span{v} (Figure 1.1 from text).
