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Subspaces of R^n

AMA1751 · Chapter overview

Subspaces of R^n

Develops span, independence, subspaces, bases, dimension, the fundamental matrix spaces, and rank.

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Lessons

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1

What can we make from the given vectors?

Understand linear combinations and span; Test span membership by solving equations; Determine whether vectors span the whole space.

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2

Are any of these vectors redundant?

Test linear independence; Write a linear dependence relation; Distinguish pivots in every row from pivots in every column.

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3

Which sets are subspaces?

Check the subspace conditions; Distinguish column space from null space; Express a homogeneous solution set as a span.

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4

How do we choose a basis for a space?

Understand bases, unique representation, and dimension; Find bases of column spaces and null spaces; Apply the same procedures to a transpose.

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5

Which vectors give a row space basis?

Find a row space basis; Explain how row operations affect row spaces and column spaces differently.

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6

Connect the chapter using rank

Use the relation between rank and null space dimension; Check restrictions imposed by matrix size; Connect rank with invertibility and solve combined questions.

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Linear Algebra · AMA1751