The concept of "image" in linear algebra

The image of a linear transformation or matrix is the span of the vectors of the linear transformation. (Think of it as what vectors you can get from applying the linear transformation or multiplying the matrix by a vector.) It can be written as Im(A).

To see why image relates to a linear transformation and a matrix, see the article on linear transformation.

For example, consider the matrix (call it A)

$ \begin{bmatrix} 1 & 0 \\ 0 & 2 \\ 0 & 1 \end{bmatrix} $

Multiplying this by a 2x1 gives a 3x1 matrix. However, regardless of what vector is chosen to multiply by, there are some vectors that can't be the result. Thus, these vectors are not in the image of A. (and thus, this is why the image matters)

The vectors that are possible belong to the span of A. In this case, the span can be represented by a "parametrized" matrix, where t and s can be any number:

$ \begin{bmatrix} 1 & 0 \\ 0 & 2 \\ 0 & 1 \end{bmatrix}* \begin{bmatrix} s \\ t\end{bmatrix} = \begin{bmatrix} s \\ 2t \\ t \end{bmatrix}=Im(A) $

A related concept is that of kernel of a matrix A.

The dimensions of the image and the kernel of A are related in the Rank Nullity Theorem


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