To prove that the subspace spanned by the column vectors of the representation matrix $A$ of a linear map $f: K^n \to K^m$ is isomorphic to the image of $f$, let us proceed step by step.

Key Concepts:

1. Representation Matrix: The matrix $A$ represents the linear map $f$, where $A \in K^{m \times n}$.
2. Column Space: The subspace spanned by the column vectors of $A$, denoted by $\text{Col}(A)$, is a subspace of $K^m$.
3. Image of $f$: The image (or range) of $f$, denoted by $\text{Im}(f)$, is the set of all vectors in $K^m$ that can be written as $f(x) = A x$ for some $x \in K^n$.

Steps of the Proof:

1. Column Space of $A$:

   • The column vectors of $A$ represent the outputs of $f$ when the basis vectors of $K^n$ are mapped under $f$.
   • For any $x \in K^n$, $A x$ is a linear combination of the columns of $A$.
   • Therefore, $\text{Im}(f) = \text{span}\{A_1, A_2, \ldots, A_n\}$, where $A_1, A_2, \ldots, A_n$ are the columns of $A$.

2. Subspace Relationship:

   • Since $\text{span}\{A_1, A_2, \ldots, A_n\} = \text{Col}(A)$, the column space of $A$ is the image of $f$.
   • Hence, $\text{Col}(A) \subseteq K^m$, and it is the same subspace as $\text{Im}(f)$.

3. Isomorphism:

   • The linear map $f: K^n \to K^m$ has a representation in $K^m$ through the column vectors of $A$.
   • The map from the basis vectors of $K^n$ to the column vectors of $A$ preserves linear dependence and independence.
   • Therefore, the column space of $A$ (which is $\text{Col}(A)$) is isomorphic to $\text{Im}(f)$.

Conclusion:

The subspace spanned by the column vectors of $A$, $\text{Col}(A)$, is indeed a subspace of $K^m$ and is isomorphic to the image of the linear map $f$, $\text{Im}(f)$. This completes the proof.

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5 Related Questions:

1. What conditions ensure that the column space of $A$ is equal to $K^m$?
2. How does the rank of $A$ relate to the dimensions of $\text{Im}(f)$?
3. What is the relationship between the null space of $A$ and the kernel of $f$?
4. How can the rank-nullity theorem be applied to connect $\text{Col}(A)$ and $\text{Im}(f)$?
5. What happens to $\text{Col}(A)$ if $A$ is a square matrix with full rank?

Tip:

To better understand the isomorphism between column space and image, practice mapping basis vectors of $K^n$ to their images in $K^m$ using simple examples.