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Chapter 3 Algebraic Properties of Linear Maps (AT)
Learning Outcomes
How can we understand linear maps algebraically?
By the end of this chapter, you should be able to...
Determine if a map between vector spaces of polynomials is linear or not.
Translate back and forth between a linear transformation of Euclidean spaces and its standard matrix, and perform related computations.
Compute a basis for the kernel and a basis for the image of a linear map, and verify that the rank-nullity theorem holds for a given linear map.
Determine if a given linear map is injective and/or surjective.
Readiness Assurance.
Before beginning this chapter, you should be able to...
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State the definition of a spanning set, and determine if a set of Euclidean vectors spans \(\IR^n\text{.}\)
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State the definition of linear independence, and determine if a set of Euclidean vectors is linearly dependent or independent.
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State the definition of a basis, and determine if a set of Euclidean vectors is a basis.
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Find a basis of the solution space to a homogeneous system of linear equations.