Duplicate Question: Show that there exist ordered bases β and γ for V and W, such that T is a diagonal matrix.

by user_hello1   Last Updated August 13, 2019 21:20 PM

Let $$\mathsf{V}$$ and $$\mathsf{W}$$ be vector spaces such that $$\dim{\mathsf{V}} = \dim{\mathsf{W}}$$, and let $$\mathsf{T: V \to W}$$ be linear. Show that there exist ordered bases $$\beta, \gamma$$ such that $$[\mathsf{T}]_\beta^\gamma$$ is a diagonal matrix.

If $$[\mathsf{T}]_\beta^\gamma$$ is diagonal, then $$\mathsf{T}(\beta) = \{\mathsf{T}(\beta_1), \dots, \mathsf{T}(\beta_n)\}$$ is linearly independent. Suppose otherwise, $$[\mathsf{T}]_\beta^\gamma$$ is diagonal but there is some linearly dependent $$\mathsf{T}(\beta_m)$$. (Informal) Then $$\mathsf{T}(\beta_m)$$ can be written in terms of the remaining elements, such that the linear matrix will be undiagonal.

$$\mathsf{T}(\beta)$$ is linearly independent if and only if $$\mathsf{T}(\beta)$$ generates $$\mathsf{W}$$ (and therefore only if $$\mathsf{T}$$ is onto). If $$\mathsf{T}(\beta)$$ is linearly independent, then it is a linearly independent set of size $$\dim{\mathsf{W}}$$. Hence it is a basis of $$\mathsf{W}$$. If $$\mathsf{T}(\beta)$$ generates $$\mathsf{W}$$, then again, given its size, it is a basis for $$\mathsf{W}$$.

By a Theorem stated in my textbook, having the same conditions as the question, $$\mathsf{T}$$ is onto if and only if $$\mathsf{T}$$ is one-one.

So it seems that if $$[\mathsf{T}]_\beta^\gamma$$ is diagonal, $$\mathsf{T}$$ must be one-one? That does not seem right.

Edit: I am aware that there is an exact duplicate Show that there exist ordered bases $\beta$ and $\gamma$ for V and W, such that T is a diagonal matrix) of the question, however I do not wish for an answer as is there provided, but only for limited guidance.

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