Math Challenges

Submissions for Problem #27

Problem #27

\[ \displaylines{\text{Suppose a linear transformation }T:\mathbb{R}^{n}\to\mathbb{R}^{n}\text{ has the property that }T\left(\mathbf{u}\right)=T\left(\mathbf{v}\right)\\ \text{ for some pair of distinct vectors }\mathbf{u},\mathbf{v}\in\mathbb{R}^{n}.\text{ Can }T\text{ map }\mathbb{R}^{n}\text{ onto }\mathbb{R}^{n}?\text{ Why or why not?}} \]

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zapwai

Solution: \( \)

Explanation:
Since the linear transformation T is not invertible, it is not onto, by the invertible matrix theorem. Or by the rank-nullity theorem, since the nullity is at least one, the rank is less than n. Or if you make a basis for the n-dimensional space that contains the vector u-v, the image is of dimension at most n-1.