• List of topics for first hour test
• Study guide and worksheet for second hour test
• List of topics for third hour test
  

Definitions
Rⁿ
unit vector
line in Rⁿ (parametric form)
hyperplane in Rⁿ  (cartesian or normal form)
linear combination of vectors
dot product of vectors
orthogonal (or perpendicular) vectors
angle between (non-zero) vectors
elementary row operations
echelon and reduced echelon matrices, pivot
rank of a matrix
singular, non-singular matrices
elementary matrices
inverses and one-sided inverses of matrices
transpose
symmetric matrix

Basic Computations
operations on scalars, vectors and matrices
matrix reduction
solving systems of linear equations
translating problems involving lines, planes, curve fitting, linear combinations and matrix equations to systems of linear equations
passing between cartesian and parametric representatios of lines and plance
projections and angles between vectors
computing inverses of matrices

Conceptual understanding
existence and uniquenes of solutions to Ax=b
good and bad features of vector and matrix operations, especially multiplication
matrices of projections, reflections and rotations in R²

Study Guide and Worksheet for Test 2

• Strive to understand the precise definitions of these terms and their geometric connotations in R² and R³; compare with examples in text and class notes.
• Learn the computational techniques of the chapter, e.g., how to use matrices to test for independence, etc, and be sure you understand why they work.
• Learn how to use the theorems of the chapter to construct your own examples, and to understand relationships among the basic concepts, e.g., that every set of 4 vectors in R³ is dependent. (The worksheet below should give you practice with this.)
• Review proof-type problems.

Fill in the following table. In each case, S refers to the set of all linear combinations of the given vectors, and T refers to the orthogonal complement of S.

Ambient Space	Vectors	Independent?	Span ambient space?	Basis for ambient space?	Dim of S	Geom. Desc. of S	Basis for T
R2	(1,2)
R2	(1,2), (1,3)
R2	(1,2), (1,3), (2,5)
R2	(1,2), (2,4), (3,6)
R2	(1,2), (2,4)
R3	(0,0,0)
R3	(1,1,1), (2,2,2)
R3	(1,1,1), (1,2,1), (1,3,1)
R3	(1,1,1), (1,1,2)
R3	(1,1,1), (1,2,1), (2,4,2)
R3	(1,1,1), (2,2,2), (4,4,4)

Fill in the blanks.
a)   Let A be an m x n matrix of rank r.  The columns of A are independent if _____.  The cols of A span Rⁿ if _____.  The cols of A are a basis for R^m if _____.
b)   A linearly independent set in R⁴ cannot have more than _____ vectors.
c)    Two distinct vectors in R⁴ which cannot together be part of a basis for R⁴ are __________.

True or False ?
a)  The vector (2,3,0,1) can be expressed as a linear combination of (1,0,0,0) and (0,1,0,1).
b)  The solution set of 2x+y-z=0 is a subspace of R³.
c)  The solution set of 2x+y-z=5 is a subspace of R³.
d)  If v₁, v₂, v₃ are dependent vectors, then there are c₁,c₂,c₃ in R so that c₁v₁+c₂v₂+c₃v₃=0 and |c₁|+|c₂|+c_3|>0.
e)  Any three vectors in R⁷ are independent.
f)   A set which spans R³ can contain 5 vectors.
g)   The set S:={(x,y,z) | x+y=z } is a subspace of R³.
h)   The set S:={(x,y,z) | x²+y²=z² } is a subspace of R³.
i)   The set S:={(x,y,z) | x²+y²+z² = 0 } is a subspace of R³.
j)   There is a set of 5 vectors in R⁷ which is independent.
k)  Any set of 4 vectors in R⁴ which is independent forms a basis for R⁴.
l)    There is a 3-dimensional subspace of R⁵.

List of Topics for Third Hour Test (typed by Ms. Stagg)

Section4.1

Projection onto a subspace V.

subspace V of Rⁿ

orthogonal complement V┴ 

V+V┴ = Rⁿ,  V∩V┴ = {0}

relation to matrices: R(A) and N(A) are orthogonal complements of each other 

ProjVb is closest member of V to b

methods for computing projections

               as in Chapter 1 when V is one-dimensional

               via bases for V, V┴

               via normal equations

               computing projection onto V┴ first

applications to data fitting

Section 4.2

orthogonal, orthonormal bases

               their advantages in computing coordinates, projections

Gram-Schmidt Process

Section 4.3: Linear Transformations 

definition

relation to matrices 

examples of linear transformations in R²: projections, rotations, reflections

examples of linear transformation on spaces of functions.

standard matrix of a linear transformation from Rn to Rm 

matrix of a linear transformation relative to a basis

kernels & images

Section 5.2 : Determinants

definition for 2x2 matrices

characterizing properties [effect of row operations]

definition for n x n matrices by above

Other properties 

               det (AB)=(det A)(det B) 

               det (AT)=det A 

               det A=0 ↔ A is singular

computation by row operations

shortcut for upper/lower triangular matrices

Section 5.3

Computation via expansion by cofactors

Sections  6.1,  6.2 & 6.3: Diagonalization 

definition of eigenvalues, eigenvectors & eigenspaces

characteristic polynomial and finding eigenvalues

eigenvectors corresponding to distinct eigenvalues are independent

finding a basis for E(λ) = null[A-λI] 

definition of diagonalizable matrix, linear transformation

criteria for diagonalizability

equations AP = PD ;  A= PDPֿ¹

               application to computing powers of A

               application to word problems