Dimension and rank - Subspace of \(\Bbb{R}^n\)

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Dimension and rank

Lessons

Notes:
Dimension of a Subspace
The dimension of a non-zero subspace SS (usually denoted as dim SS), is the # of vectors in any basis for SS. Since the null space and column space is a subspace, we can find their dimensions.

Note: Dimension of the column space = rank

Finding the Rank of a matrix:
1. Find the basis for the column space
2. Count the # of vectors in the basis. That is the rank.
Shortcut: Count the # of pivots in the matrix

Finding the dimensions of the null space:
1. Find the basis for the null space
2. Count the # of vectors in the basis. That is the dimension.
Shortcut: Count the # of free variables in the matrix.

The Rank Theorem
If a matrix AA has nn columns, then rank A+A+ dim N(A)=nN(A) = n.
  • Intro Lesson
    Dimension and Rank Overview:
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Dimension and rank

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