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Math 20F. Final Exam December 7, 2005
Read each question carefully, and answer each question completely. Show all of your work. No credit will be given for unsupported answers. Write your solutions clearly and leg
Print Name: Math 20F. Midterm Exam 1 October 17, 2005
Student Number:
Read each question carefully, and answer each question completely. Show all of your work. No credit will be given for unsupported answers. Write your solutions clearly and legibly. No c
Math 20F Linear Algebra
Lecture 1
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A primer on Linear Algebra
Slide 1
Remarks on the course.
Overview of Linear Algebra.
Systems of linear equations.
(Row approach) (Sec. 1.1).
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Linear Algebra is the study of Vector Spaces
Example of vector
Math 20F Linear Algebra
Lecture 4
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A matrix is a function
Slide 1
Review: Column picture.
Matrix equation Ax = b.
A matrix is a linear function.
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The column picture is essential for linear algebra
2x1 x2 = 0,
x1 + 2x2 = 3.
Slide 2
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x1
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Math 20F Linear Algebra
Lecture 12
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Change of basis
Slide 1
Review: Components of a vector in a basis.
Change of basis.
Review: Midterm 1.
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Any vector can be decomposed in an unique way
in terms of a basis vectors
Slide 2
Theorem 1 Let V be
Math 20F Linear Algebra
Lecture 15
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Linear Transformations
Domain, range, and null spaces.
Slide 1
Injective and surjective transformations.
Bijections and the inverse.
Nullity + Rank Theorem.
Components in a basis: Matrices.
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Linear tran
Math 20F Linear Algebra
Lecture 18
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Determinants, n n
Review: The 3 3 case.
Slide 1
Determinants n n.
(Expansions by rows and columns.
Relation with Gauss elimination matrices: Properties.)
Formula for the inverse matrix.
Cramers rule.
(To solve
Math 20F Linear Algebra
Lecture 6
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Matrix operations
Review:
Slide 1
Linear combinations of matrices.
Multiplication of matrices.
Gauss elimination using matrix multiplication.
Inverse matrix.
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Matrices are a new type of vectors!
Because
Math 20F Linear Algebra
Lecture 9
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The beginning of Linear Algebra
Review
Slide 1
Broad overview.
Main results (so far).
Examples.
The need of abstraction: Vector Space.
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Main results so far including an invertible matrix
Theorem 1 Let A
Math 20F Linear Algebra
Lecture 20
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On determinants and eigenvalues
Slide 1
Review: eigenvalues and eigenvectors.
Eigenspaces.
Characteristic equation.
Multiplicity of eigenvalues.
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The eigenvectors of a matrix determine directions
where t
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Math 20F. Midterm Exam 2 November 21, 2005
Read each question carefully, and answer each question completely. Show all of your work. No credit will be given for unsupported answers. Write your solutions clearly an