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'''4.1 Vectors in the Plane and in 3-Space''' | '''4.1 Vectors in the Plane and in 3-Space''' | ||
| − | Basic definitions of what a vector and a coordinate system is (see book). I am under the impression that you have had enough math to know what these are. | + | Basic definitions of what a vector and a coordinate system is (see book). I am under the impression that you have had enough math to know what these are. |
<br> | <br> | ||
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A '''real vector space''' is a set V of elements on which we have two operations + and ∙ defined with the following properties: | A '''real vector space''' is a set V of elements on which we have two operations + and ∙ defined with the following properties: | ||
<blockquote> (a) If u and v are any elements in V, then u + v is in V. We say that V is '''closed''' under the operation +<br> 1. u + v = v + u for all u, v in V<br> 2. u + (v + w) = (u + v) + w for all u, v, w in V<br> 3. There exists an element 0 in V such that u + 0 = 0 + u = u for any u in V<br> 4. For each u in V there exists an element –u in V such that u + -u = -u + u = 0<br> (b) If u is any element in V and c is any real number, then c ∙ u is in V<br> 1. c ∙ (u + v) = c ∙ u + c ∙ v for any u, v in V and any real number c<br> 2. (c + d) ∙ u = c ∙ u + d ∙ u for any u in V and any real numbers c and d<br> 3. c ∙ (d ∙ u) = (cd) ∙ u for any u in V and any real numbers c and d<br> 4. 1 ∙ u = u for any u in V </blockquote> | <blockquote> (a) If u and v are any elements in V, then u + v is in V. We say that V is '''closed''' under the operation +<br> 1. u + v = v + u for all u, v in V<br> 2. u + (v + w) = (u + v) + w for all u, v, w in V<br> 3. There exists an element 0 in V such that u + 0 = 0 + u = u for any u in V<br> 4. For each u in V there exists an element –u in V such that u + -u = -u + u = 0<br> (b) If u is any element in V and c is any real number, then c ∙ u is in V<br> 1. c ∙ (u + v) = c ∙ u + c ∙ v for any u, v in V and any real number c<br> 2. (c + d) ∙ u = c ∙ u + d ∙ u for any u in V and any real numbers c and d<br> 3. c ∙ (d ∙ u) = (cd) ∙ u for any u in V and any real numbers c and d<br> 4. 1 ∙ u = u for any u in V </blockquote> | ||
| − | <br>'''4.3 Subspaces''' | + | The operation "+" is called '''vector addition '''and the operation "∙" is '''scalar multiplication'''. <br>''' '''''<b> </b>'' |
| + | |||
| + | ''''' '''Examples'': | ||
| + | <blockquote> | ||
| + | # Let R<sup>n</sup> be the set of all 1 x n matrices [a<sub>1 </sub>a<sub>2</sub> ··· a<sub>n</sub>], where we define "+" by | ||
| + | [a<sub>1</sub> a<sub>2</sub> ··· a<sub>n</sub>] + [b<sub>1</sub> b<sub>2 </sub>··· b<sub>n</sub>] = [a<sub>1</sub>+ b<sub>1 </sub>a<sub>2</sub> + b<sub>2</sub> ··· a<sub>n</sub> + b<sub>n</sub>] and we define "·" by c · [a<sub>1</sub> a<sub>2</sub> ··· a<sub>n</sub>] = [ca<sub>1</sub> ca<sub>2</sub> ··· ca<sub>n</sub>] </blockquote> | ||
| + | 2. The same principles can be applied to polynomials | ||
| + | |||
| + | p(t) = a<sub>n</sub>t<sup>n</sup> + a<sub>n-1</sub>t<sup>n-1</sup> + ··· + a<sub>1</sub>t + a<sub>0</sub> | ||
| + | |||
| + | <sub></sub> q(t) = b<sub>n</sub>t<sup>n</sup> + b<sub>n-1</sub>t<sup>n-1</sup> + ··· + b<sub>1</sub>t + b<sub>0</sub> | ||
| + | |||
| + | |||
| + | |||
| + | p(t) + q(t) = (a<sub>n</sub>+b<sub>n</sub>)t<sup>n </sup>+ (a<sub>n-1</sub> + b<sub>n-1</sub>)t<sup>n-1</sup> + ··· + (a<sub>1</sub> + b<sub>1</sub>)t + (a<sub>0</sub> + b<sub>0</sub>) | ||
| + | |||
| + | c · p(t) = (ca<sub>n</sub>)t<sup>n</sup> + (ca<sub>n-1</sub>)t<sup>n-1</sup> + ··· + (ca<sub>1</sub>)t + (ca<sub>0</sub>) | ||
| + | |||
| + | <br> | ||
| + | <blockquote></blockquote> | ||
| + | '''4.3 Subspaces''' | ||
A subspace is a “mini” vector space that satisfies all of the properties mentioned in section 4.2. An easy way to test if something is a subspace is to see if it satisfies the addition and scalar multiplication properties. | A subspace is a “mini” vector space that satisfies all of the properties mentioned in section 4.2. An easy way to test if something is a subspace is to see if it satisfies the addition and scalar multiplication properties. | ||
Revision as of 09:22, 16 December 2010
Elementary Linear Algebra Chapter 4: Real Vector Spaces
Welcome!
Note: This page is based on the fourth chapter in Elementary Linear Algebra with Applications (Ninth Edition) by Bernard Kolman and David R Hill.
4.1 Vectors in the Plane and in 3-Space
Basic definitions of what a vector and a coordinate system is (see book). I am under the impression that you have had enough math to know what these are.
4.2 Vector Spaces
A real vector space is a set V of elements on which we have two operations + and ∙ defined with the following properties:
(a) If u and v are any elements in V, then u + v is in V. We say that V is closed under the operation +
1. u + v = v + u for all u, v in V
2. u + (v + w) = (u + v) + w for all u, v, w in V
3. There exists an element 0 in V such that u + 0 = 0 + u = u for any u in V
4. For each u in V there exists an element –u in V such that u + -u = -u + u = 0
(b) If u is any element in V and c is any real number, then c ∙ u is in V
1. c ∙ (u + v) = c ∙ u + c ∙ v for any u, v in V and any real number c
2. (c + d) ∙ u = c ∙ u + d ∙ u for any u in V and any real numbers c and d
3. c ∙ (d ∙ u) = (cd) ∙ u for any u in V and any real numbers c and d
4. 1 ∙ u = u for any u in V
The operation "+" is called vector addition and the operation "∙" is scalar multiplication.
Examples:
[a1 a2 ··· an] + [b1 b2 ··· bn] = [a1+ b1 a2 + b2 ··· an + bn] and we define "·" by c · [a1 a2 ··· an] = [ca1 ca2 ··· can]
- Let Rn be the set of all 1 x n matrices [a1 a2 ··· an], where we define "+" by
2. The same principles can be applied to polynomials
p(t) = antn + an-1tn-1 + ··· + a1t + a0
q(t) = bntn + bn-1tn-1 + ··· + b1t + b0
p(t) + q(t) = (an+bn)tn + (an-1 + bn-1)tn-1 + ··· + (a1 + b1)t + (a0 + b0)
c · p(t) = (can)tn + (can-1)tn-1 + ··· + (ca1)t + (ca0)
4.3 Subspaces
A subspace is a “mini” vector space that satisfies all of the properties mentioned in section 4.2. An easy way to test if something is a subspace is to see if it satisfies the addition and scalar multiplication properties.
4.4 Span
Let S be a set of vectors in a vector space V. If every vector in V is a linear combination of the vectors in S, then the set S is said to span V, or V is spanned by the set S; that is, span S = V.
4.5 Linear Independence
Linear Independence is when all vectors in a set of vectors are unique. So if there are two vectors in a set that are a combination of other vectors in the the set, then the set is not linear independent.
4.6 Basis and Dimension
The vectors in a vector space V are said to form a basis for V if they:
(1) span V
(2) linear independent
The dimension of a nonzero vector space V is the number of vectors in a basis for V. We often write dim V for the dimension of V. We also define the dimension of the trivial vector space {0} to be zero.
4.7 Homogeneous Systems
4.9 Rank of a Matrix
• If A is an m × n matrix, then rank A + nullity A =n
• A is nonsingular if and only if rank A = n
• If A is an n × n matrix, then rank A = n if and only if det(A) ≠ 0
