MAT 1341: REVIEW II
1. Projections and Cross Product
1.1. Projections.
Denition 1.1. Given a vector , the rectangular (or perpendicular or orthogonal) comu
ponents are two vectors 1 and 2 such that = 1 + 2 and 2 (1 and are
u
u
u
u
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u1
u u
u2
perpendi
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MAT 1341: VECTOR SPACES I
1. Vector spaces
A vector space is an algebraic structure formed by a collection of vectors. For example, the set of (cartesian) vectors in R2 is a vector space, i.e., The 2-dimensional space
R2 = cfw_[a, b]| a, b R (by identifyi
MAT 1341: VECTOR SPACES IV
1. Basis and Dimension
1.1. Basis.
Denition 1.1. Let V be a vector space. A basis is a set of linearly independent vectors
that spans V .
Example 1.2.
(1) Let W = Rv be a subspace of V , where v is a nonzero vector. Then cfw_v i
MAT 1341: VECTOR SPACES III
1. Linear Independence
Note that a spanning set (i.e., the set of all linear combinations of given vectors) forms a
subspace even though there may be dierent expressions for the same vector. For instance, as
1
R2 = spancfw_e1 ,
MAT 1341: VECTOR SPACES II
1. Subspaces
Denition 1.1. A subset W of a vector space V is called a subspace of V if W is itself a
vector space under the addition and the scalar multiplication dened in V .
Consider the line L : [x, y, z] = t[1, 3, 4] (i.e.,
MAT 1341: REVIEW III
1. Planes
1.1. Equations of Planes. As we see in the previous section, a line (in any dimensional
space) is uniquely determined either by two points or by a nonzero vector and a point.
Similarly, a plane in 3-dimensional space is uniq
MAT 1341: REVIEW I
1. Complex Numbers
A real number is a point on an innitely long number line. More precisely, the set of real
numbers, denoted by R, is dened by completing (i.e., adding limits of sequences of rational
numbers to the eld) the rational nu