##### Exercise2

Determine whether the following statements are true or false and provide a justification for your response.

1. A matrix transformation $$T:\real^4\to\real^5$$ is defined by $$T(\xvec) = A\xvec$$ where $$A$$ is a $$4\times5$$ matrix.

2. If $$T:\real^3\to\real^2$$ is a matrix transformation, then there are infinitely many vectors $$\xvec$$ such that $$T(\xvec) = \zerovec\text{.}$$

3. If $$T:\real^2\to\real^3$$ is a matrix transformation, then it is possible that every equation $$T(\xvec) = \bvec$$ has a solution for every vector $$\bvec\text{.}$$

4. If $$T:\real^n\to\real^m$$ is a matrix transformation, then the equation $$T(\xvec) = \zerovec$$ always has a solution.

5. If $$T:\real^n\to\real^m$$ is a matrix transformation and $$\vvec$$ and $$\wvec$$ two vectors in $$\real^n\text{,}$$ then the vectors $$T(\vvec + t\wvec)$$ form a line in $$\real^m\text{.}$$

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