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Vectors and Vector Algebra
Reviews vectors in R^n, vector addition, scalar multiplication, and their algebraic properties.
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AMA1751 · Chapter overview
Subspaces of R^n
Develops span, independence, subspaces, bases, dimension, the fundamental matrix spaces, and rank.
Lessons
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Linear Combinations and Span
Determines whether vectors belong to a span and translates spanning questions into matrix equations.
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3
Linear Independence and Dependence
Tests linear independence using homogeneous equations, pivots, and dependence relations.
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4
Subspaces and Subspace Proofs
Checks subspace conditions and writes concise closure proofs or counterexamples.
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5
Null Space and Column Space
Interprets Nul A and Col A as subspaces and tests membership in each space.
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6
Bases for the Fundamental Subspaces
Finds bases for null, column, and row spaces from row-reduction data.
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7
Basis, Dimension and Basis Theorems
Builds and reduces spanning sets, identifies bases, and applies dimension and basis theorems.
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8
Row Space, Rank and Rank-Nullity
Connects row space, column space, pivots, rank, nullity, and the invertible matrix theorem.
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9
Normal Equations and Projection Decomposition
Synthesizes A^T A, normal equations, projection matrices, and the decomposition between Col(A) and Nul(A^T).
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Linear Algebra · AMA1751