Chapter 6.3, Problem 1TFR

For Questions (a)-(f), decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem from the text. If false, provide an example, illustration, or brief explanation of why the statement is false.

(a) If $T:V\to W$ is a linear transformation and $W$ is finite-dimensional, then $\dim \left[\text{Ker}\left(T\right)\right]+\dim \left[\text{Rng}\left(T\right)\right]=\dim \left[W\right]$

(b) If $T:P_4\left(ℝ\right)\to {ℝ}^7$ is a linear transformation, then $\text{Ker}\left(T\right)$ must be at least two-dimensional.

(c) If $T:{ℝ}^n\to {ℝ}^m$ is a linear transformation with matrix $A$, then $\text{Rng}\left(T\right)$ is the solution set to the homogeneous linear system $Ax=0$.

(d) The range of a linear transformation $T:V\to W$ is a subspace of $V$.

(e) If $T:M_{23}\left(ℝ\right)\to P_7\left(ℝ\right)$ is a linear transformation with $T\left[\begin{array}{ccc}1& 1& 1\\ 0& 0& 0\end{array}\right]=0$ and $T\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]=0$, then $\text{Rng}\left(T\right)$ is at most four-dimensional.

(f) If $T:{ℝ}^n\to {ℝ}^m$ is a linear transformation with matrix $A$ then $\text{Rng}\left(T\right)$ is the column space of $A$.