Overview of Week 5
Math 208 Section A, October 25, 2021
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Reading Assignment:
- Section 3.2, due Friday 10/15/21.
- Section 3.3, due Friday 10/29/21.
- Section 3.2, due Friday 10/29/21.
- Sections 4.1 and 4.2, Due Friday 11/05/21
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- Section 3.2, due 10/25/21
- Section 3.3, due 10/27/21
- Section 4.1, due 11/01/21
- Section 4.2, due 11/03/21
- Section 4.3, due 11/08/21
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Conceptual Problems:
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Vocabulary List:
- Chapter 3:
- linear transformtions, domain, codomain, range (or image), null space (or kernel)
- "one-to-one" and "onto" linear transformations
- Unifying Theorem, Version 2
- matrix addition and scalar multiplication
- matrix multiplication
- the identity matrix
- matrix algebra
- matrix transpose
- composition of linear transformations and matrix multiplication
- powers of a matrix
- diagonal, upper triangular, and lower triangular matrices
- elementary matrices and elementary row operations
- partitioned matrices
- the inverse of a linear transformation and inverse matrices
- computing matrix inverses
- Unifying Theorem, Version 3
- Chapter 4:
- Subspace
- The range and null space of a matrix as subspace.
- the range and kernel of a linear transformation as subspaces.
- How to obtain the matrix associated with a linear transformation.
- The Unifying Theorem: Versions 4, 5, and 6
- A basis for a subspace.
- The dimension of a subspace.
- The row space of a matrix.
- The column space of a matrix.
- The rank and nullity of a matrix.
- The Rank Plus Nullity Theorem.
- Coordinate vectors with respect to a given basis.
- The standard basis.
- Change of basis matrices, and change of basis formulas.
- Change of basis in a subspace.
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Key Concepts:
- Chapter 3
- the relationship between linear transformations and matrices
- properties of "one-to-one" and "onto" linear transformations
- all the versions of the Unifying Theorem
- matrix algebra
- composition of linear transformations and matrix multiplication
- matrix inverses and their computation
- elementary row operations as matrix products
- Chapter 4
- subspaces and vectors that span them
- basis and dimension of a subspaces
- range and null space of a matrix
- range and kernel of linear transformation
- row and column space of a matrix
- rank and nullity of a matrix and the Rank Plus Nullity Theorem
- change of bases
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Skills to Master:
- Chapter 3
- testing if a matrix is one-to-one or onto
- computing an elementary matrix
- computing a matrix inverse
- applying the Unifying Theorem
- Chapter 4
- Checking to see if a set of vectors is a subspace.
- Computing bases for the range and null space of a matrix.
- Computing bases for the range and kernel of a linear transformation.
- Computing a basis for a subspace and determining it dimension.
- Computing bases for the row and column spaces of a matrix.
- Computing the dimensions of the row and column spaces of a matrix.
- Computing the rank and nullity of a matrix.
- Applying the Rank Plus Nullity Theorem.
- Computing a change of basis matrix.