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. True or false: 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. True or false: 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