REVIEW SHEET FOR THE
MIDTERM
The midterm
will cover the material we saw in class and studied in homework, section 1.1
through 3.3 (but only those that were covered!). Review definitions and
theorems covered in class, as well as those highlighted in the book. Below is
the list of most fundamental concepts that we studied as well as questions
related to them.
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:
-
Be
able to solve a linear system. Make use of back substitution (checking your answer)
to prevent losing points because of your arithmetic.
-
Be
able to determine whether a given vector is a linear combination of a given set
of vectors
-
Be
able to determine whether a given vector is in the Span of a given collection
of vectors
-
Be
able to determine whether a given set of vectors is independent
The introductory
“Be able” is going to be skipped starting at this point
-
Determine
the standard matrix of a linear transformation described geometrically
-
Determine
whether a given transformation is linear.
CHAPTER II
Definitions:
-
Inverse
of a matrix
-
Subspace
of Rn
-
Column
space and Null space of a matrix
-
Basis
-
Coordinates
with respect to a given basis
-
Dimension
-
Rank
Concepts:
-
Matrix
operations and their properties
-
Correspondence
between matrix multiplication and composition of linear transformations
-
Elementary
matrices
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
-
Skills
-
Computations
of determinants: explicit formula for 2x2, 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 for matrices 2x2