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Preface

One goal of this Preface can be achieved right away. You need to know about the video lectures for MIT's Linear Algebra course Math 18.06. Those videos go with this book, and they are part of MIT's OpenCourseWare. The direct links to linear algebra are

https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/ https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/

On YouTube those lectures are at https://ocw.mit.edu/1806videos and @1806scvideos

The first link brings the original lectures from the dawn of OpenCourseWare. Problem solutions by graduate students (really good) and also a short introduction to linear algebra were added to the new 2011 lectures. And the course today has a new start—the crucial ideas of linear independence and the column space of a matrix have moved near the front.

I would like to tell you about those ideas in this Preface.

Start with two column vectors a1a₁ and a2a₂. They can have three components each so they correspond to points in 3-dimensional space. The picture needs a center point which locates the zero vector:

a1=[231],a2=[142],zero vector=[000]\boldsymbol{a}_1 = \begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix}, \quad \boldsymbol{a}_2 = \begin{bmatrix} 1 \\ 4 \\ 2 \end{bmatrix}, \quad \text{zero vector} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}

The vectors are drawn on this 2-dimensional page. But we all have practice in visualizing three-dimensional pictures. Here are a₁, a₂, 2a₁, and the vector sum a₁ + a₂.

a2=[142],a1+a2=[373]\boldsymbol{a}_2 = \begin{bmatrix} 1 \\ 4 \\ 2 \end{bmatrix}, \quad \boldsymbol{a}_1+\boldsymbol{a}_2 = \begin{bmatrix} 3 \\ 7 \\ 3 \end{bmatrix} 0=[000],a1=[231],2a1=[462]\boldsymbol{0} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}, \quad \boldsymbol{a}_1 = \begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix}, \quad 2\boldsymbol{a}_1 = \begin{bmatrix} 4 \\ 6 \\ 2 \end{bmatrix}