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MIT 18.06SC Strang Problem Set 1.1

(:description Exercises:Strang folios 8-12)

1. Describe geometrically (line, plane, or all of {$\mathbb R ^3 $}) all linear combinations of

{$$ \text{(a) } \begin{bmatrix} 1 \cr 2 \cr 3 \end{bmatrix} \text{ and } \begin{bmatrix} 3 \cr 6 \cr 9 \end{bmatrix} \quad \text{(b) } \begin{bmatrix} 1 \cr 0 \cr 0 \end{bmatrix} \text{ and } \begin{bmatrix} 0 \cr 2 \cr 3 \end{bmatrix}$$} {$$ \text{(c) } \begin{bmatrix} 2 \cr 0 \cr 0 \end{bmatrix} \text{ and } \begin{bmatrix} 0 \cr 2 \cr 2 \end{bmatrix} \text{ and } \begin{bmatrix} 2 \cr 2 \cr 3 \end{bmatrix} $$}

2. Draw {$ \mathbf v = \begin{bmatrix} 4 \cr 1 \end{bmatrix} \text{ and } \mathbf w = \left[\begin{array}{r} -2 \cr 2 \end{array}\right] \text{ and } \mathbf v + \mathbf w \text{ and } \mathbf v - \mathbf w $} in a single xy plane.

3. If {$ \mathbf v + \mathbf w = \begin{bmatrix} 5 \cr 1 \end{bmatrix} \text{ and } \mathbf v - \mathbf w = \begin{bmatrix} 1 \cr 5 \end{bmatrix}$}, compute and draw {$ \mathbf v \text{ and } \mathbf w$}.

4. From {$ \mathbf v = \begin{bmatrix} 2 \cr 1 \end{bmatrix} \text{ and } \mathbf w = \begin{bmatrix} 1 \cr 2 \end{bmatrix}$}, find the components of {$ 3 \mathbf v + \mathbf w \text{ and } c \mathbf v + d \mathbf w$}.

5. Compute {$ \mathbf u + \mathbf v + \mathbf w \text{ and } 2 \mathbf u + 2 \mathbf v + \mathbf w$}. How do you know {$\mathbf u ,\ \mathbf v ,\ \mathbf w$} lie in a plane?

{$$ \textbf{In a plane }\quad \mathbf u = \begin{bmatrix} 1 \cr 2 \cr 3 \end{bmatrix},\ \mathbf v = \left[\begin{array}{r} -3 \cr 1 \cr -2 \end{array}\right], \ \mathbf w = \left[\begin{array}{r} 2 \cr -3 \cr -1 \end{array}\right] $$}

6. Every combination of {$ \mathbf v = (1, -2, 1) \text{ and } \mathbf w = (0, 1, -1) $} has components that add to _______. Find {$ c \text{ and } d \text{ so that } c \mathbf v + d \mathbf w = (3,3, -6).

7. In the xy plane mark all nine of these linear combinations:

{$$ c \begin{bmatrix} 2 \cr 1 \end{bmatrix} + d \begin{bmatrix} 2 \cr 1 \end{bmatrix} \quad \text{ with } c = 0,\,1,\,2\ \text{ and } d = 0,\,1,\,2 $$}


Sources:

File:Linear subspaces with shading.svg by Alksentrs Wikimedia. Creative Commons Attribution-Share Alike 3.0 Unported license. The three-dimensional Euclidean space R3 is a vector space, and lines and planes passing through the origin are vector subspaces in R3.

Strang, Gilbert. Introduction to Linear Algebra. 4th ed. Wellesley, MA: Wellesley-Cambridge Press, February 2009. [SG]

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