Section 1.3: Vector Equations
- Reading Goals:
Understand algebraic and graphical representation of vectors; know how to convert from vector equations to
linear
systems of equations to augmented matrices; understand linear combinations and the span of a set of vectors.
- Required Problems: # 11, 24, 26, 32.
- Recommended Problems: # 9, 12, 13, 21, 23.
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Section 1.4: The Matrix Equation Ax = b
- Reading Goals:
Be able to write a vector equation as a matrix equation; determine whether a matrix equation is consistent
and list equivalent conditions; know basic properties of matrix-vector products.
- Required Problems: # 9, 15, 22, 24.
- Recommended Problems: # 10, 14, 23, 31.
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Section 1.5: Solution Sets of Linear Systems
- Reading Goals:
Find solution sets to homogeneous systems of linear equations and relate to nonhomogeneous linear systems;
write solution sets in parametric vector form.
- Required Problems: # 5, 15, 24, 40.
- Recommended Problems: # 6, 16, 23, 26, 29.
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Section 1.7: Linear Independence
- Reading Goals:
Understand linear dependence and independence; understand the relationship between
dependence of the columns of a matrix and the number of solutions to the corresponding
matrix equation.
- Required Problems: # 5, 10, 22.
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Section 1.8: Introduction to Linear Transformations
- Reading Goals:
Know the properties of a linear transformation; recognize geometric properties of
some linear transformations.
- Required Problems: # 11, 19, 22, 30.
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Section 1.9: The Matrix of a Linear Transformation
- Reading Goals:
Produce a matrix equation for a given linear transformation; understand one-to-one and onto
transformations and their relationship with the number of solutions.
- Required Problems: # 7, 24, 35.
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Section 2.1: Matrix Operations
- Reading Goals:
Know how to multiply matrices (when the product is defined), properties of matrix multiplication,
and the transpose of a matrix.
- Required Problems: # 2, 16, 25.
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Section 2.2: The Inverse of a Matrix
- Reading Goals:
Know how to solve a system of linear equations using the inverse of a matrix,
understand the relationship between row operations and elementary matrices,
be able to compute the inverse of a matrix (learn the formula for the inverse of a 2-by-2 matrix).
- Required Problems: # 7, 10.
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Section 2.3: Characterizations of Invertible Matrices
- Reading Goals:
Know and be able to apply the Invertible Matrix Theorem.
- Required Problems: # 2, 8, 12.
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Section 3.1: Introduction to Determinants
- Reading Goals:
Understand the relationship between the determinant and invertibility of a matrix,
be able to compute the determinant.
- Required Problems: # 10, 40.
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Section 3.2: Properties of Determinants
- Reading Goals:
Understand how row operations change the value of the determinant,
know how to quickly computer the determinant of a triangular matrix,
simplify the determinant of a product of matrices.
- Required Problems: #12, 26, 28.
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Section 4.1: Vector Spaces and Subspaces
- Reading Goals:
Know the definition of a vector space and subspaces and examples.
- Required Problems: #8, 12, 20, 24.
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Section 4.2: Null Spaces, Column Spaces, Linear Transformations
- Reading Goals:
Be able to computer the null space and column space of a matrix,
understand how they are related and why they are important.
- Required Problems: #5, 15, 26, 29.
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Section 4.3: Linearly Independent Sets; Bases
- Reading Goals:
Know the definition of basis and examples; be
able to check whether a given set of vectors forms a basis; be able to find a basis for a given space;
know how to use the Spanning Set Theorem.
- Required Problems: #9, 13, 22.
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Section 4.4: Coordinate Systems
- Reading Goals:
Understand the Unique Representation Theorem; find the change-of-coordinates matrix from the standard basis of Rn to a vector space with basis B and use it to convert from standard coordinates into
B-coordinates and back again.
- Required Problems: #7, 16, 32.
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Section 4.5: Dimension of a Vector Space
- Reading Goals:
Know the definition of dimension; understand the Basis Theorem and how it simplifies verification of the conditions
for a basis; be able to find the dimensions of the Null Space and Column Space of a matrix and how they are related.
- Required Problems: #14, 20, 25.
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Section 5.2: The Characteristic Equation
- Reading Goals:
Understand the relationship between the eigenvalues of a matrix and the characteristic polynomial; be able to compute
characteristic polynomials; know the definition of similar matrices.
- Required Problems: #7, 10, 15, 22.
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Section 5.3: Diagonalization
- Reading Goals:
Know how the factorization A = PDP-1 can be used to calculate a formula for Ak;
be able to diagonalize a matrix.
- Required Problems: #4, 11.
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Section 6.1: Orthogonality, Inner Products, and Length
- Reading Goals:
Know how to compute the inner product of vectors and how it relates to the length; know how to compute the
distance between two vectors; know the definition of orthogonality, how to determine when vectors are orthogonal,
and how it relates to the Pythagorean theorem.
- Required Problems: #20, 24.
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