REVIEW SHEET FOR THE FINAL
CHAPTER I.
Definitions:
-
consistent and
inconsistent linear systems;
-
pivoting positions of a
matrix
-
linear combination
-
Span
-
linear independence/ linear dependence
-
linear transformation
-
one-to-one
-
onto
Concepts:
- Systems of linear equations. Equivalent
forms of systems of linear equations: matrix form, vector form.
- Row reduction algorithm, row equivalent
matrices, echelon and reduced echelon forms
- Homogeneous linear systems
- Parametric vector form of a solution
- Standard matrix of a linear transformation
Theorems:
-
Theorem 4
-
Theorem 6
-
Theorem 11
-
Theorem 12
Skills:
- Solving
linear systems by row reduction. Make use of back substitution (checking your
answer) to prevent losing points because of your arithmetic.
-
Determine whether a
given vector is a linear combination of a given set of vectors
-
Determine whether a
given vector is in the Span of a given collection of vectors
-
Determine whether a
given set of vectors is independent
-
Determine the standard
matrix of a linear transformation described geometrically
-
Determine whether a
given transformation is linear.
CHAPTER II
Definitions:
-
Inverse of a matrix
-
Column space and Null
space of a matrix
-
Transpose of a matrix
- Elementary
matrices
Concepts:
-
Matrix operations and
their properties
-
Correspondence between
matrix multiplication and composition of linear transformations
Theorems:
-
Theorem 3
-
Theorem 4
-
Theorem 8 ( and all the expansions to it that we added
later)
-
Theorem 13
Skills:
-
Adding, multiplying,
transposing, inverting matrices
-
Expressing row
operations in terms of multiplication by elementary matrices
-
For a 2x2 matrix, know
two ways to find the inverse
-
Be very familiar with
various equivalent descriptions of invertible matrices (i.e. Theorem 8 +
additions to it)
-
Computations of rank,
dimension of the Null space, basis of column space and Null space
CHAPTER III
Concepts
-
Determinant
-
Determinant as a volume
Theorems
-
Theorem 2
-
Theorem 4
-
Theorem 6
- Theorem
8
-
Theorem 10
Skills
-
Computations of
determinants: explicit formulae for 2x2, 3x3, decomposition with respect to a
row/column, row reduction
-
Determinants of
elementary matrices
-
Computations of area
(in R2) and volume (in R3) using determinants
-
Cramer’s rule
- Computing
inverse matrices using the Inverse formula
- Characterization
of an invertible matrix in terms of the determinant
CHAPTER IV
Definitions
-
Vector
space
-
Subspace
-
Linear
transformation
-
Kernel
and Image of a linear transformation
-
Basis
- Coordinates
relative to a basis
- Dimension
of a vector space
- Row
space of a matrix
Concepts
-
Change
of coordinates matrix
-
Standard
basis of Rn
-
Change
of basis
-
Rank
as dimension of the row space and the column space
Theorems
-
Theorem
7
-
Theorem
* from class (properties of the coordinate map) (with proof)
-
Theorem
9
-
Theorem
10
-
Theorem
11
-
Theorem
12 (the basis theorem)
- Theorem
13 (the rank theorem) (with proof)
Skills
-
Determine
bases and dimension for Column, Row and Null space of a matrix
-
Determine
whether a given set of vectors form a basis
-
Compute
coordinates with respect to given basis,
compute new coordinates out of old and vice versa
-
Applications
of the rank theorem