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Linear Algebra II

See rank, span, and kernel account for what a linear map keeps and loses.

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Columns define the directions an output can use. A matrix combines its columns according to the input coordinates, so column independence controls the dimension of the image.

subspaces of a linear mapR² → R² · A = [ 2 1 ; 1 2 ]
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Change the columns
Column space and kernel of a two by two matrixThe columns are 2, 1 and 1, 2. Their rank is 2, their nullity is 0, and the sample input maps to 3, 3.uvA[1,1]column space · all of R²0
column ucolumn vsample output
Matrix AA = [ 2 1 ; 1 2 ]
Sample mapA[1, 1]ᵀ = [3, 3]ᵀ
Null vector[0, 0]ᵀ → [0, 0]ᵀ
rank2
nullity0
determinant3
rank + nullity2 + 0 = 2
Column spaceall of R²

Two independent columns span both output dimensions.

Kerneltrivial

Only the zero input maps to zero.

Dimension check2 + 0 = 2

Input dimension = image dimension + lost dimension.

The two columns point in different directions, so their linear combinations fill the plane. Rank 2 means the image keeps both dimensions.

Rank measures how many independent output directions survive. Nullity measures how many input directions disappear. The rank–nullity theorem keeps the accounting exact: every lost input dimension appears in the kernel.