cises 32–36, column vectors are written as rows, such as x

QUESTION:

In Exercises 32–36, column vectors are written as rows, such as \(\mathbf{x}=\left(x_{1}, x_{2}\right)\) and T (x) is written as \(T\left(x_{1}, x_{2}\right)\).

Let \(T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\) be a linear transformation. Show that if T maps two linearly independent vectors onto a linearly depend set, then the equation T(x)=0 has a nontrivial solution. [Hint: Suppose u and v in \(\mathbb{R}^{n}\) are linearly independent and yet T(u) and T(v) are linearly dependent. Then \(c_{1} T(\mathbf{u})+c_{2} T(\mathbf{v})=\mathbf{0}\) for some weights \(c_1\) and \(c_2\), not both zero. Use this equation.]