consider notations and representations for vectors,
learn two, basic arithmetic operations for vectors,
use these to define an important structure.
By the end of this section you should know the following terms and be able to recognize their concept when you see it.
linear combination
span
You should also be able to perform the following calculations.
vector arithmetic including addition, scalar multiplication,
matrix arithmetic including addition, scalar multiplication, vector multiplication, matrix multiplication
Subsection1.2.1Interpretation
An important aspect of mathematics is discovering when details can be ignored. In applications like physics or calculus there is a difference between a vector and a point. However, we know that the arithmetic is the same. Below are three notations which will have limited if any impact on linear algebra. The last two will be illustrated later in this section.
A point \((1,-3,-2)\text{.}\)
A row vector \([1,-3,-2]\text{.}\)
A column vector \(\left[\begin{array}{r} 1 \\ -3 \\ -2 \end{array}\right]\text{.}\)
Subsection1.2.2Vector Arithmetic
This checkpoint uses your past knowledge and illustrates the next concept.
Checkpoint1.2.1.
Use your knowledge of vector arithmetic from previous classes to perform each arithmetic operation indicated. Also sketch each vector. These sketches will help visualize vector arithmetic. You may use the 3D Calculator in Geogebra 1 to do this.
In Geogebra first create a point by typing (1,-3,-2) or similar for each vector. It will be drawn as a point. Then select the Vector tool and click on the start point (perhaps (0,0,0)) and endpoint you just created.
(a)
\(\left[1,-3,-2\right]+\left[2,7,3\right]= \)
(b)
\(2\left[1,-3,-2\right]=\)
(c)
\(-5\left[2,7,3\right]= \)
(d)
\(2\left[1,-3,-2\right]+(-5)\left[2,7,3\right]=\)
(e)
\(2\left[1,-3,-2\right]-5\left[2,7,3\right]= \)
(f)
Compare and contrast the last two results.
Subsection1.2.3Span
This experiment will help you discover the geometric interpretation of the concept of a span.
In Geogebra you can create a point by typing (1,-3,-2) or similar.
(a)
Sketch each of the following as points.
\(\displaystyle 0[1,-3,-2]+2[2,7,3]\)
\(\displaystyle 2[1,-3,-2]+2[2,7,3]\)
\(\displaystyle 1[1,-3,-2]+0[2,7,3]\)
\(\displaystyle 1[1,-3,-2]+2[2,7,3]\)
\(\displaystyle -1[1,-3,-2]+2[2,7,3]\)
(b)
Describe the collection of all the points \(a[1,-3,-2]+b[2,7,3],\text{.}\)
The following terms will be used extensively in this course.
Definition1.2.2.Linear Combination.
The sum of scaled vectors, for example \(1[1,-3,-2]+1[2,7,3]+3[-4,5,-1], \) is called a linear combination of those vectors.
Definition1.2.3.Span.
The set of all vectors (points) that can be obtained as a linear combination of a set of vectors, for example \(a[1,-3,-2]+b[2,7,3]+c[-4,5,-1] \) for all real numbers \(a,b,c,\) is called the span of those vectors.
Subsection1.2.4Matrix Arithmetic
Matrices will be used often as examples of vectors. Here we describe arithmetic on them as will be needed to treat them as vectors.
The following illustrates multiplying a matrix by a vector on the right.
Complete the following calculations. You will need to extrapolate the technique from the previous operations. This is an exercise in generalizing a technique.
