Linear Algebra: Vector Spaces

Visualizing Vector Spaces

So far we have only been dealing with vector spaces as abstract concepts. However, a picture's worth a thousand words, so we take to drawing out vectors.

The easiest vector space to visualize is $\mathbb{R}^2$, the real plane. The whole vector space is depicted as the plane, and vectors within the vector space are drawn as rays emanating from the origin and pointing towards their coordinates. While we could draw vectors as isolated points, the ray visualization emphasizes the fact that vectors have both a magnitude and a direction, which form the key interpretation of vectors in subjects such as geometry, physics, and computer graphics. For example, the vectors $u=[3, 1]$, $v=[-5, -5]$, and $w=[-2, -2]$ are depicted below.

Vectors in R^2

Vector addition is visually pleasing to perform in the plane. If you draw one vector, then draw the second emanating from the tip of the first vector rather than the origin, it will point to the sum of the two vectors. The next figure depicts the sum of $u+v$.

Vector addition

Scalar multiplication is simple to visualize, as a scalar multiple $c$ of a vector $v$ is another vector pointing in the direction of $v$ but $c$ times as long. In $\mathbb{R}^n$, if $c$ is negative, then the vector points in the opposite direction. The next figure shows $w$ scaled by a factor of $-2$.

Scalar multiplication

The problems below will give you some practice visualizing vector arithmetic.

Problems

  1. Draw and label the following vectors in $\mathbb{R}^2$:

    • $x = [2, 4]$

    • $y = [3, -2]$

    • $z = [-4, -2]$

    Drawing of x, y, and z in R^2
    Show Answer

  2. Let $a=[1, 2]$ and $b=[0, 1]$ be vectors in $\mathbb{R}^2$. Draw the following sum of vectors by drawing each scaled vector tail to tip with the previous one:

    1. Calculate $a + 2b -3a -3b + a$

    2. Starting from the origin, draw the each scaled vector term in the above sum tail to tip with the previous term.

    1. $ a + 2b -3a -3b + a = [1,2] + 2[0,1] -3[1,2] -3[0,1] + [1,2] \\ a + 2b -3a -3b + a = [1,2] + [0,2] + [-3,-6] + [0,-3] + [1,2] \\ a + 2b -3a -3b + a = [1 + 0 - 3 + 0 + 1, 2 + 2 - 6 - 3 + 2] \\ a + 2b -3a -3b + a = [-1, -3] \\ $

    2.  

    Vector sum drawn tail to tip: a + 2b -3a -3b + a
    Show Answer

  3. Consider $\mathbb{C}$ as a vector space. Multiplication of complex numbers is different from multiplication of real numbers. For complex numbers $x = a + bi$ and $y = c + di$, their product is $xy = ac - bd + adi + bci$. This definition leads to the fun fact that $i^2 = -1$.

    We can visualize $\mathbb{C}$ as a coordinate plane, where the real part is plotted horizontally and the imaginary part is plotted vertically. Thus, a complex number $a + bi$ would be displayed just as the vector $[a, b] \in \mathbb{R}^2$ would be displayed.

    1. Let $z = 4 + 4i$. Calculate $iz$, $i^2z$, and $i^3z$

    2. Draw $z$, $iz$, $i^2z$, and $i^3z$ to show that multiplication by $i$ is equivalent to rotation in the plane by $\pi/2$ radians (or 90 degrees).

    1. $iz = i(4+4i) = 4i + 4i^2 = -4 + 4i

      $i^2z = i(iz) = i(-4 + 4i) = -4i + 4i^2 = -4 - 4i$

      $i^3z = i(i^2z) = i(-4 - 4i) = -4i - 4i^2 = 4 - 4i$

    2.  

    Multiplication by $i$ is equivalent to rotation by pi/2 radians
    Show Answer