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Section 4.2: Subspaces
Def: A subspace of a vector space V is a subset W of
V that is itself a vector space using the same addition
and scalar multiplication as in V .
Theorem: Suppose that V is a vector space. A nonempty subset W of V is a subspace of V
Section 4.4: Coordinates and Basis
Def: Let V be a vector space and S = cfw_v1 , . . . , vn a set
of vectors in V . S is a basis for V if
1. S is a linearly independent set, and
2. S spans V.
Note: A set of column vectors cfw_v
Section 4.6: Change of Basis
Problem: Let V be a two dimensional vector space and
B = cfw_u1 , u2 , B = cfw_u1 , u2 two bases for V. If v is a vector
in V and (v)B = (k1 , k2 ) nd (v)B .
Theorem If we change the basis for a vector space
Section 4.5: Dimension
Def: A nonzero vector space V is nite dimensional
if there exists a nite basis S = cfw_v1 , . . . , vn of V . V
is innite dimensional otherwise. We regard the zero
vector space as nite dimensional.
Theorem All bases for a
Section 4.7: Row Space, Column Space, and Nullspace
For an m n matrix A:
The subspace of Rn spanned by the row vectors of A
is called the row space of A and is denoted row(A).
The subspace of Rm spanned by the column vectors
of A is called the colu
Section 4.8: Rank, Nullity and the Fundamental Matrix Spaces
For any matrix A
dim(row(A) = dim(col(A).
Def. The common dimension of the the row space and
column space of a matrix A is called the rank of A,
denoted rank(A). The dimension of
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ECHELON FORMS . GAT]SSIAN
A rectangular matrix that has the fotlowing
1. The leftmost nonzero entry in each row is a 1
called the leadine 1
2. Alt nonzero ro\rys are above any rows of all
3. In any two successive
Practice Exam 1
1. Find a basis for the row space, column space, and null space of the matrix given
4 0 7
1 5 2 2
1 4 0 3
1 1 2 2
1 0 0 1
Solution. rref (A) = 0 1 0 1 . Thus
Strategy for Testing Series: Solutions
1. Since (5)n = (1/5)n , this is a geometric series. Because |1/5| < 1, it converges.
2. Since n2 < n2 + 6n = n(n + 6) for all n 0, we have
n(n + 6)
1/n2 converges (its a p-series with p = 2 > 1),
MA322 021 Midterm 2 - 7/16/07
Name: Directions: Please print your name clearly. This is 60 minute test and is worth 20% of your nal grade. There are 90 points possible. Calculators are not permitted on this exam. You must show all of your work. Answers wi
Math 214 Solutions to Assignment #6
20. Identify the curve r = tan sec by nding a Cartesian equation for the curve.
x = r cos = cos = x/r and tan = y/x. So
r = tan sec =
= 2 = y = x2 ,
i.e., the curve is a parabola opening upw
Theorem 4.4.1 (Uniqueness of Basis Representation) If S = cfw_v1 , v2 , . . . , vn is a basis for a vector space V, then every vector v in V can be expressed in the form v = c1 v1 + c2 v2 + . . . + cn vn in exactly one way. Proof: Assume there is another
Section 4.3: Linear Independence
Defninition: A set of vectors cfw_v1 , v2 , . . . , vp in a vector
space V is said to be linearly independent if the vector
c1 v1 + c2 v2 + + cp vp = 0
has only the trivial solution c1 = 0, . . . , cp = 0.
Section 4.1: Real Vector Spaces
We can think of a real vector space in general, as a
collection of objects that behave like vectors do in Rn .
Def. A real vector space is a nonempty set V of objects, called vectors, on which there are two operations,
Theorem: Lagrange's Identity
w 3 . Then v w
v = (v 1 , v 2 ,v 3 ) and w
= (w 1 ,w 2 , w3) . It is a bit tedious, but a
Proof: Just work out both sides letting
v and w
. Then v w
MATH 2004 Homework Solution
Homework 4 Model Solution
Section 13.2 13.4.
13.2.3 Let r(t) = ht 2, t2 + 1i.
(a) Sketch the plane curve with the given vector equation.
x = t 2, y = t2 + 1 t = x + 2 y = (x + 2)2 + 1
y = x2 + 4x + 5
(b) Find r0 (
1. Use cofactor expansion along a suitable row or column to compute det(A) if
(No marks if cofactor expansion is not used for the (44)-determinant as well for all resulting
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