4.5 Basis and Dimension
- Structure of Vector Spaces
Recognize bases in the vector spaces Rn, Pn,
and Mm,n.
Test a basis in a vector space
Find the dimension of a vector space
Definition of Basis
A set of vectors S = cfw_ v1, v2, , vn in a vector
spac
Sea 2~S. C/mmcfariamfevg of md Wfas.
IL. 4% A 5e. a your: w mam Tim 4%
cfw_bl/awj gaws m gin/wad, 57* A jrm 4
17% WW1! m 6% a/ ue. 9% Al/ lm.
a. A is fhWtxluz
b A 7; MW 7mm6m1 40 #19. mm TWP/t5] Mix.
C. has n [Dis/of fvslens.
0!. AK 6 Ans 91.x, m m! gown
H
L.
-7
f0? WA 1;) In
:8 v T3
1X, 1 SXZ
3 7< I ' 7 X2.
2 S
1 _ I
km A @03st
71m. 69, (A0" = 4
6) (A8)" = B"/.\' a m was; (,f
_r _ , T mm fhWr'tiLL MFrCQS
C) (A ) z 64") is [we/mtg, M m
in verse is #19. [won/act?
a: cfw_Mr cfw_WW [(4
14/12, NW
lo 0 0,
1/2.).
E7 ;S Ct a] ) , an
24; gram cfw_07, m, 3; 19M %$a% ac K Scotland
by V? V.
gloom/E7I - is u oa/amffpm of all Vecerj
.6mf W Le, cur/CW I"l W 70m
0,7, 7L 6sz 4p + 6,74,
WM Cl , , m
Z;
7.
Geo mail-K paserTf-hem z
_ 3 ._ _,
If 72,7 air-L mam mac-667:9
Math 207 Exam 1 Name: H M r 1/5
*>l<>l<>l<*>l<*>I<*>l<*>l<>k*>k>l<*>l<*>K>l<>l0l<>l<>l<>t>k>l<*>k><*>l<>k*$>l<*"q<*>k*>l<*>l<*at*>l<>l<*>l<>l<*>k*>k>k>k*>l<*>l<>l<*>k>k>k*>l<>l<>k>k>l<>l<*>k$*>k*>k*
Please Show your work on the paper to get full credit.
*
Exam 4 Concepts and Skills (Covering Section 5.4, 6.1 - 6.3, and 7.1 - 7.2)
Sect. 5.4 Find the projection of a vector onto a subspace; solve the least
squares problem
Sec. 6.1 Find images and preimages; determine whether a given function
a linear transfor
5.3 Orthonormal Bases:
Gram-Schmidt process
Show that a set of vectors is orthogonal
Represent a vector relative to an orthonormal
basis
Apply Gram-Schmidt orthonormalization
process
Review: Basis for a vector space V
S = cfw_ v1, v2, , vn is called a
Chapter 6 Linear Transformations
In Chapter 4, we built up vector spaces with two
operations (addition and scalar multiplication)
satisfying 10 properties. We explored inside a vector
space (subspaces, spanning sets, linear independence,
bases, dimension,
Chapter 5 Inner Product Spaces
In this chapter we will enrich vector spaces
by adding another operation, called Inner
Product. Through it we will extend some
concepts in Rn, such as length, distance, angle,
orthogonality, to more general vector
spaces.
5.
Chapter 3 Determinants (Find Determinants)
3.1 Determinant of a Matrix
Find the determinant of a 2 x 2 matrix
Find the minors and cofactors of a matrix
Use expansion by cofactors to find the
determinant of a matrix
Find the determinant of a triangular
Math 207 FINAL EXAM Name (printed): I- 5,; [H
*
Show your work. Your answers must be justied to get full credit.
*>k*
l 4
1. (15 pts) Given A = < 1 3 >
a. Find A1 by adjoining the identity and using GaussJordan elimination.
L; (IQ'Io)RWQzw(/+ 0
0/1!
b.
4.7 Coordinates and Change of Basis
Find a coordinate matrix relative to a basis
in Rn.
Find the transition matrix from basis B to
B in Rn.
Represent coordinates in general
n-dimensional spaces.
Coordinates Relative to a Basis
Let B = cfw_v1, v2, vn be
Review of Section 6.1 and 6.2
Chapter 6 - Functions from a vector space onto a
vector space
Section 6.1 Introduced to the special type of
functions linear transformations
functions satisfying the linearity
T(c1v1 + c2v2) = c1T(v1) + c2T(v2)
Section 6.2 T
6.4 Transition Matrix and Similarity
Introduction to the problem
Let T: V V be a linear transformation. B and B
are two bases for V. A is the matrix for T relative
to B and A is the matrix for T relative to B
([T(v)]B = A[v]B and [T(v)]B = A[v]B )
Q. What
cfw_reL vMaLCL. 9%er /.7,
Aim CZ '
A we of mm? Wham 2e"
I:S /i if
7 - AM W5.
Xp\7>,+X7.VL 4" + 0
MS W 6r/M/ (XI=XL: Xf=0).
The 96 75 Maw/Z 0&fo if f/wc
vs6 way/$5 C, Ca, Cf , M157 n/ zero, melt
M; )eWM if m 92% 72 s
[,W/ IM.
I a) 4L 7:
7/7,:[2] VI'[ 3
Math 207
Lesson 7: Properties of determinants
Section 3.3
24 pts.
Show all work to receive full credit. Clearly indicate your answers. Erase or cross out all work you do not want considered.
Each problem is worth 4 points.
2 1
3
1 1 and B = 2
0 2
4
1 1
2
Math 207
Lesson 6: Computing determinants
Sections 3.1, 3.2
24 pts.
Show all work to receive full credit. Clearly indicate your answers. Erase or cross out all work you do not want considered.
Each problem is worth 4 points.
1. Determine all minors Mij an
Math 207
Lesson 8: Vector operations, vector spaces, and vector subspaces
Sections 4.1, 4.2, 4.3
32 pts.
Show all work to receive full credit. Clearly indicate your answers. Erase or cross out all work you do not want considered.
Each problem is worth 4 p
Math 207
Lesson 11: Kernel and range
Sections 6.1, 6.2
24 pts.
Show all work to receive full credit. Clearly indicate your answers. Erase or cross out all work you do not want considered.
Each problem is worth 4 points.
1. The transformation T : R2 R2 , d
Math 207
Lesson 4: Matrix inverses and LU factorization
Sections 2.3, 2.4
32 pts.
Show all work to receive full credit. Clearly indicate your answers. Erase or cross out all work you do not want considered.
Each problem is worth 4 points.
1. Use the formu
Math 207
Lesson 1: Solution sets of linear systems
Section 1.1
24 pts.
Show all work to receive full credit. Clearly indicate your answers. Erase or cross out all work you do not want considered.
Each problem is worth 4 points.
1. Graph the system of line
Math 207
Lesson 2: Gaussian elimination
Section 1.2
24 pts.
Show all work to receive full credit. Clearly indicate your answers. Erase or cross out all work you do not want considered.
Each problem is worth 4 points.
1. Identify the elementary row operati
Math 207
Lesson 3: Matrix arithmetic
Sections 2.1, 2.2
32 pts.
Show all work to receive full credit. Clearly indicate your answers. Erase or cross out all work you do not want considered.
Each problem is worth 4 points.
1 0
1. Compute 2A 3B for the matric
Math 207
Lesson 12: The matrix of a linear transformation
Sections 6.3
24 pts.
Show all work to receive full credit. Clearly indicate your answers. Erase or cross out all work you do not want considered.
Each problem is worth 4 points.
1. Find the standar
Math 207
Lesson 13: Eigenvalues, eigenvectors, and diagonalizable matrices
Sections 7.1, 7.2
24 pts.
Show all work to receive full credit. Clearly indicate your answers. Erase or cross out all work you do not want considered.
Each problem is worth 4 point
Math 207
Lesson 9: Linear independence, bases, and rank
Sections 4.4, 4.5, 4.6
32 pts.
Show all work to receive full credit. Clearly indicate your answers. Erase or cross out all work you do not want considered.
Each problem is worth 4 points.
1. Determin