Exercise 7

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

1. Given two vectors $\vvec$ and $\wvec$, the vector $2\vvec$ is a linear combination of $\vvec$ and $\wvec$.

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

3. Suppose $\vvec_1,\vvec_2,\ldots,\vvec_n$ is a collection of $m$-dimensional vectors and that the matrix $\left[\begin{array}{rrrr} \vvec_1 & \vvec_2 & \ldots & \vvec_n \end{array}\right]$ 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. It is possible to find two 3-dimensional vectors $\vvec_1$ and $\vvec_2$ such that every 3-dimensional vector can be written as a linear combination of $\vvec_1$ and $\vvec_2$.