Notes on linear algebra (Monday 17th October. 2016, 23:10) page 185
. If A has a row lled with zeroes, then det A = 0.
o If we add a multiple of some row of A to another row, then det A does not
chang
Notes on linear algebra (Monday 17th October, 2016, 23:10) page 207
(by (158). In other words, there exist some A1, A2,. . ., M 6 1R satisfying 2 = A1791 +
szg + . . * + Milk. Fix these A1,L2, a . .,.
Notes on linear algebra (Monday 17th October. 2016, 23:10) page 205
of IR. (Recall that IR stands for the vector space IRW1 of all column vectors of
size n.) Then, Ker A is a subspace of IR.
Propositi
Notes on linear algebra (Monday 17th October, 2016, 23:10) page 191
(thus, again, entry by entry), and 6 defined by
6 = (0,0,0,.).
(It makes sense to think of infinite sequences as 1 x ecu-matrices; t
Notes on linear algebra (Monday 17th October, 2016, 23:10) page 204
other words, It has the form (a,0,2a + '0 + 1)T for some a,v 6 IR. Consider these
a, I).
Now, h = (a, 0, 2a -| v + 1)T. Hence, It ha
Notes on linear algebra (Monday 17th October, 2016, 23:10) page 179
3. Assume now that none of the values b_k+1,bn_k+2,.,bn is nonzero.
Thus, all of the values bn_k+1, bn_k+2, . . ., b are zero. Hence
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Proposition 4.4. Property (j) in Denition 4.2 follows from properties (g) and (i).
Thus, we could omit property (j) from the definit
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o The vector space IR (from Example 4.8) can be identied with the vector
space lRcfw_1.
:- The vector space C (from Example 4.8) can
Notes on linear algebra (Monday 17th October; 2016, 23:10) page 187
(f) We have 2L(v+w) = 2w+2tw for all A E R. v E V and w E V. (This is
called left distributivity".)
(g) We have (ip) "a = 2L (pa) fo
Denition 1. Let n be a positive integer. The set IR. is dened by
:rtl.r;s.2,.,.on ER
An element _ in R is called an navector or simply a Vector; and (11,112. . . . ,on
art
are the components of the ve
(Can you prove this?) 0
Denition 2. The span of two vectors v1 and v2, written span(v1, v2), is the set
of all iinear combinations of v1 and v2. That is; it is the set
Sp(V1,Vg) = cfw_x : x = Alvl + A
Obviously, this is a line through (4, %, 0) parallel to 1 . Hence a parametric vector form of
6
the line is
3: 4 7
x: y = % +A 1 , forAER.
z 0 *6
METHOD 2. Set each term to A. That is
:1: 4 _ 2g + 3 _
6 3 3
10 5 5
As 4 = 2 2 , the span is the line 1: = A 2 for some A E R. O
14 7 7
16 8 8
The span of two non-zero vectors is either a plane or a line through the origin. If one of the
vectors, say v1,
a,"
9
x:(mm<2>+<$>=<2>+A<r>-
Thus a parametric vector form for the line is
x: (0)+)\(1) for/MEIR.
d or
You should think of this as saying that the equation 3; = ms: + d represents a line through the p
3
Example 4. Find the lengths of a = (_:) and b = 2
6
4
SOLUTION. \a| = J4 + 16 = N5; \bl = J1 + 9 + 4 + 36 + 1 = 1/66. 0
Denition 4. The distance between two points A and B with position vectors
in R
Notes on linear algebra (Monday 17th October, 2016, 23:10) page 201
single equation x1 x2 | 2x3 2 0, we now have to keep track of the two equations
x1 2 2x2 and 2x2 2 3x3. Let me only show one part of
Notes on linear algebra (Monday 17th October. 2016, 23:10) page 186
3.27. <TODO> The rest
I TODO 3.172. (ii) SageMath examples.
I TODO 3.173. (*) What holds over rings, what over elds.
4. <TODO> Vecto
Notes on linear algebra (Monday 17th October. 2016. 23:10) page 200
of A). Thus. 03 lies in A (since 0 0 + 2 - 0 = 0 holds). In other words, A contains
the zero vector.
Proof that A is closed under ad
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(here, we have renamed the indices a and o as x and y). (The reason for rewriting
D this way was to get rid of the letters a and o; i
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(Let me dwell on the precise meaning of the equalities (154), (155) and (156).
The sign l appears three times in (154), and each tim
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1. Third version (formal, uses Section 3.21): Let
dEtA : E (1)UA1,U'(1)A2,T(2) ' ' ' Amalia);
aESn
where 5. denotes the set of all pe
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Denition 4.28. Let V be a vector space. Let o1, o2, . . . , ok be nitely many vectors
in V.
(a) A linear combination of '01, 02,. .
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(c) We have AU 6 U for all A 6 IR and U E 11.
Note that condition (b) is often put into words as follows: The subset U is
closed und
Notes on linear algebra (Monday 17th October, 2016, 23:10) page 183
TODO 3.170. Let me give three versions of the definition of det A for all a:
I First version (informal and sloppy): Set
['11 2 + + +
Notes on linear algebra (Monday 17th October. 2016, 23:10) page 196
q q
(The expression 2 bok has to be read as E (192%).)
k=p k=p
Proposition 4.19. Let V be a vector space. Let p and a be two integer
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4.5. <DRAFT> Examples and constructions of subspaces
4.5.1. cfw_3 and V
Examples of subspaces, and general ways to construct subspac
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of R3. Then, C is not a subspace of R3.
(1) Let D be the subset
cfw_(a,0,.2'a+v)T | am 6 IR
of R3. (Note that the letter 0 stands fo
Notes on linear algebra (Monday 17th October, 2016, 23:10) page 190
Example 4.6. Let n E IN and m E N. The set of all a X tit-matrices with real
entries shall be denoted by IRW. This set lRWI is a vec
Notes on linear algebra (Monday 17th October, 2016, 23:10) page 193
I 106)=2(x+3)2x3+2=2+0x+0x2+13x3+12x4+2x5foranx 61R.)
Example 4.13. The set of all constant functions from ]R to ]R is a vector spac
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4.3. <DRAFT> (*) The summation sign for vectors
In Section 2.9,. we have introduced the summation sign 2 for sums of numbers;
in Sec
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:- [LaNaSc16, Chapter 8] does the definitions and the basic properties really
well. I highly recommend it.
o [Heffer16, Chapter Four