quartz/content/vault/Subspaces.md

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#math/linear_algebra

definition

A ==subspace== of \rm I\!R is any collection S in \mathbb{R}^{n} such that: ^1652733454769

1. The zero vector 0 is in S.
2. If \mathbf{u} and \mathbf{v} are in S, then \mathbf{u}+\mathbf{v} is in S. ( S is closed under vector addition.)
3. If \mathbf{u} is in S and c is a scalar, then c \mathbf{u} is in S. ( S is closed under scalar multiplication.)

homogeneous

All spans of homogeneous systems are subspaces. They all go through 0.

null space

#card Null space describes wherever the matrix solutions equal 0. Usually you find a vector that, when it multiplies the matrix, the result is always zero. When the span of that vector*matrix combo is all zeroes, that span is a null space. ^1652736387934

Basis of a Subspace

#card/reverse A set of vectors that:

• are linearly independent
• span the subspace Basically the most concise description of the subspace. ^1652736951914

Bases of \mathbb{R}^{n}

standard basis

\mathbf{e}_{1}=\left[\begin{array}{c}
1 \\
0 \\
\vdots \\
0
\end{array}\right], \mathbf{e}_{2}=\left[\begin{array}{c}
0 \\
1 \\
\vdots \\
0
\end{array}\right], \ldots, \mathbf{e}_{n}=\left[\begin{array}{c}
0 \\
0 \\
\vdots \\
1
\end{array}\right]

are called the standard basis for \mathbb{R}^{n}.

other bases of \mathbb{R}^{n}

The columns of an invertible nXm matrix are linearly independent and span \mathbb{R}^{n}.

row space of matrix

Row reduce until the 0 rows come out. Every nonzero row found after that is independent and together, span that subspace. That means they're the basis.

basis for null(matrix)

basis for null(A) is just the solution to the homogenous matrix for A. Set each column equal to zero, and the vectors that multiply those columns such that the sum equals the zero vector is the basis for null(A).

• every basis for a space has the same number of elements.
• any set in a subspace with more vectors than the basis is linearly dependent.
• the name for that number of vectors/elements is the Dimension of space S

random

nullity: The nullity of a matrix is determined by the difference between the order and rank of the matrix. The rank of a matrix is the number of linearly independent row or column vectors of a matrix. If n is the order of the square matrix A, then the nullity of A is given by n – r. !