Prove that img A ⊇img A^2. More generally, prove img A ⊇img(A B) for every compatible matrix B.

Step 1: Understand the notation and definitions.u000Au002D $u005Coperatorname{img} A$ refers to the image of

Prove that img A ⊇img A^2. More generally, prove img A...

Key Concepts

Image (Range) of a Matrix

The image of a matrix, also known as the range of the corresponding linear transformation, is the collection of all possible output vectors produced by applying the matrix to vectors from its domain. This concept is fundamental in understanding how linear transformations map between vector spaces and in determining the subspace that the outputs form.

Matrix Multiplication as Composition of Linear Transformations

Matrix multiplication can be viewed as the composition of linear transformations. When one matrix multiplies another, it is equivalent to applying one transformation after the other. This interpretation is essential when considering how the range of a composite transformation is related to the range of the initial transformation.

Subspace Inclusion

Subspace inclusion involves showing that every element of one subspace (in this case, the image of a product of matrices) is also an element of another subspace (the image of a single matrix). This concept is a key tool in linear algebra, particularly when demonstrating relationships between different ranges or kernels of linear maps.

Proof by Element-wise Argument

A common technique in linear algebra is to prove set inclusion by taking an arbitrary element from one set and demonstrating it belongs to another. Using this method helps structure rigorous proofs, such as showing that any vector in the image of a product of matrices can also be obtained by the action of one of the matrices alone.

Prove that $\operatorname{img} A \supseteq \operatorname{img} A^2$. More generally, prove $\operatorname{img} A \supseteq \operatorname{img}(A B)$ for every compatible matrix $B$.

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