Exercise 5

Determine whether the following statements are true or false and provide a justification for your response. Throughout, we will assume that the matrix $A$ has columns $\vvec_1,\vvec_2,\ldots,\vvec_n$; that is, $$A = \left[\begin{array}{rrrr} \vvec_1&\vvec_2&\ldots&\vvec_n \end{array}\right]$$.

1. If the equation $A\xvec = \bvec$ is consistent, then $\bvec$ is in $\laspan{\vvec_1,\vvec_2,\ldots,\vvec_n}$.

2. The equation $A\xvec = \vvec_1$ is consistent.

3. If $\vvec_1$, $\vvec_2$, $\vvec_3$, and $\vvec_4$ are vectors in $\real^3$, then their span is $\real^3$.

4. If $\bvec$ is a linear combination of $\vvec_1, \vvec_2,\ldots,\vvec_n$, then $\bvec$ is in $\laspan{\vvec_1,\vvec_2,\ldots,\vvec_n}$.

5. If $A$ is an $8032\times 427$ matrix, then the span of the columns of $A$ is a set of vectors in $\real^{427}$.