##### Exercise7

Provide a justification for your response to the following statements or questions.

1. True of false: Given two vectors $$\vvec$$ and $$\wvec\text{,}$$ the vector $$2\vvec$$ is a linear combination of $$\vvec$$ and $$\wvec\text{.}$$

2. Suppose $$\vvec_1,\vvec_2,\ldots,\vvec_n$$ is a collection of $$m$$-dimensional vectors and that the matrix $$\left[\begin{array}{rrrr} \vvec_1 \amp \vvec_2 \amp \ldots \amp \vvec_n \end{array}\right]$$ has a pivot position in every row. If $$\bvec$$ is any $$m$$-dimensional vector, then $$\bvec$$ can be written as a linear combination of $$\vvec_1,\vvec_2,\ldots,\vvec_n\text{.}$$

3. Suppose $$\vvec_1,\vvec_2,\ldots,\vvec_n$$ is a collection of $$m$$-dimensional vectors and that the matrix $$\left[\begin{array}{rrrr} \vvec_1 \amp \vvec_2 \amp \ldots \amp \vvec_n \end{array}\right]$$ has a pivot position in every row and every column. If $$\bvec$$ is any $$m$$-dimensional vector, then $$\bvec$$ can be written as a linear combination of $$\vvec_1,\vvec_2,\ldots,\vvec_n$$ in exactly one way.

4. Suppose we are working with 3-dimensional vectors. Is it possible to find two vectors $$\vvec_1$$ and $$\vvec_2$$ such that every 3-dimensional vector can be written as a linear combination of $$\vvec_1$$ and $$\vvec_2\text{?}$$

in-context