Math 2B – Final Review
Revised: August 5th, 2015 – Hiep Ton
MATH-2B Review for Final Exam
1. Main facts about system of linear equations, including rewriting the system as a matrix
equation; basis of the column space, the row space, and the null space of a matrix.
2. Construction of the basis of the orthogonal complement for the subspace spanned by the
given system of vectors; the orthogonal projection of the given vector on the given
subspace.
3. The least square solution of the given inconsistent system of linear equations using the
normal system. The geometric interpretation of the solution.
4. Solving a system of linear equations using LU-decomposition.
5. A matrix transformation, its matrix and finding the image of the given vector under the
transformation.
6. Linear transformation in 2-space and its matrix.
7. A reduction of a quadratic form to principal axes and its geometric interpretation.
8. Solving the Initial Value Problem for the linear system of ordinary differential equations
using methods of linear algebra.
9. Checking that the given system of vectors form a basis for the given vector space. Finding
the coordinates of the given vector with respect to the given basis; QR-decomposition of the
given matrix.
10. Markov’s chains and its applications.
You need to know main definitions, theorems (without proofs), and main procedures (algorithms).
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Math 2B – Final Review
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1. Main facts about system of linear equations, including rewriting the system as a matrix
equation; basis of the column space, the row space, and the null space of a matrix.
SECTION 1.1:
SECTION 4.7: Row / Column / Null space
Example 3 – finding the basis
NOTE: Finding basis by reducing A to ref (without doing any transpose).
Parameterized for all 0s row.
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Math 2B – Final Review
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Example 7 – Inconsistent row and column space from given matrices
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Example 8 – Inconsistent row and column space from given vectors
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Math 2B – Final Review
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Example 9 – Consistent row and column space from given matrices
NOTE: if question asks for consisting from original given matrices, then need to use transpose
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Math 2B – Final Review
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2. Construction of the basis of the orthogonal complement for the subspace spanned by the
given system of vectors; the orthogonal projection of the given vector on the given subspace.
Reference: Projection formula (in Calculus series)
Focus on Gram-Schmidt process
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3. The least square solution of the given inconsistent system of linear equations using the
normal system. The geometric interpretation of the solution.
SECTION 6.1: Inner Product
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Math 2B – Final Review
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SECTION 6.4: Least Square
SECTION 6.4: The Geometric Interpretation of the Solution
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Math 2B – Final Review
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4. Solving a system of linear equations using LU-decomposition.
SECTION 9.1:
Example: Finding L and U through E and its inverse
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Math 2B – Final Review
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Example: Finding L and U directly
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5. A matrix transformation, its matrix and finding the image of the given vector under the
transformation.
SECTION 4.9: Reflection Operator
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SECTION 4.9: Projection Operator
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SECTION 4.9: Rotation Operator
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SECTION 4.9: Dilations and Contractions
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SECTION 4.9: Expansion and Compressions
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SECTION 4.9: Shears
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6. Linear transformation in 2-space and its matrix.
SECTION 4.9: Matrix Transformations from Rn to Rm
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SECTION 10.10:
Problem 4 – From Project part 1
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7. A reduction of a quadratic form to principal axes and its geometric interpretation.
SECTION 7.3 :
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8. Solving the Initial Value Problem for the linear system of ordinary differential equations using
methods of linear algebra.
SECTION 5.4: Differential Equation
Note: If the given A matrix has already been diagonalized, then no need to find D since A,
itself is the diagonal matrix.
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Example 2 – The given matrix A is NOT diagonal
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Math 2B – Final Review
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9. Checking that the given system of vectors form a basis for the given vector space. Finding the
coordinates of the given vector with respect to the given basis; QR-decomposition of the given
matrix.
SECTION 6.3: The Gram-Schmidt Process
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SECTION 6.3: QR – Decomposition
Example 9 – Make sure using q’s as orthonormal from the orthogonal set after Gram-Schmidt
Process and also check if the set is orthogonal before processing QR
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10. Markov’s chains and its applications.
SECTION 4.12:
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Math 2B – Final Review
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NOTE: Highly recommended look at your Project part 1.
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