Lesson: Vector Spaces Over \( \mathbb{C} \)

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Transcript

Now that we know how to add and multiply complex numbers, we can look at vectors from Cn. Just as in Rn, a vector in Cn will be an ordered list of n complex numbers, and we will define addition and scalar multiplication on vectors componentwise.

So the vector space Cn is defined to be the set Cn equal to the sets of all lists [z1 through zn] of complex numbers, with addition of vectors defined by the vector [z1 through zn] + the vector [w1 through wn] will equal the vector [(z1 + w1) through (zn + wn)], and scalar multiplication of vectors is defined by s[z1 through zn] = [sz1 through szn] for all s in C. So again, notice that our scalars are now from the complex numbers, not the real numbers.

We also extend the notion of complex conjugate to vectors in Cn. The complex conjugate of the vector z = [z1 through zn] in Cn is defined to be vector z-conjugate, which equals [(the conjugate of z1) through (the conjugate of zn)].

Let’s look at some examples of operations in the complex vector space. So if we looked at the vector [2 + 3i; i; -1 – 5i] and add it to the vector [7 – 5i; 6; 4 + i], then we will get the vector whose first component is the sum of our first components, the second component is the sum of the second components, and the third component is the sum of the third components, and performing these calculations, we see that we get the vector [9 – 2i; 6 + i; 3 – 4i].

We can look at the scalar multiple of (3 – i) times the vector [1 + i; 1 – 2i; 5 – i]. So in this case, we’re going to be looking at (3 – i) times our first component, (3 – i) times our second component, and (3 – i) times our third component. So we can perform those calculations, and we’ll see that the answer is the vector [4 + 2i; 1 – 4i; 14 – 8i].

Of course, combining scalar multiplication and vector addition can get us linear combinations of vectors in Cn. So we could look at ((2 + 4i) times the vector [-3 – 3i; 5 + 4i]) – ((3i) times the vector [4; 7 + i]). Now, I like to perform my scalar multiplication first, so first I’ll get the vector [-6 – 6i – 12i – 12(i-squared); 10 + 8i + 20i + 16(i-squared)] plus the vector [-12i; -21i – 3(i-squared)]. And then you can go ahead and add your first components, and add your second components, and you’ll see that you end up with the vector [6 – 30i; -27 + 25i].

And lastly, we’ll note that if the vector z is [2 + 3i; 1 – 5i; 2; -3i], then the conjugate of the vector z is the vector whose first component is (2 + 3i)-conjugate, whose second component is (1 – 5i)-conjugate, the third component is the conjugate of 2, and the fourth component is the conjugate of (-3i), and that this will equal [2 – 3i; 1 + 5i; 2; 3i].

As always, you’ll see that our definitions of addition and scalar multiplication satisfy the vector space properties, although this time, we’re going to replace R with C, so let’s look at the definition.

A vector space over C is a set V, together with an operation of addition (usually denoted x + y for any x and y in V) and an operation of scalar multiplication (usually denoted sx for any x in V and s in C), such that for any x, y, and z in V, and s and t in C, we have all of the following properties:

1. V1 is that x + y is in V. Using words, we’ll say that our vector space is closed under addition.
2. V2 says that (x + y)-quantity + z must equal x + the quantity (y + z), or in words, that addition is associative.
3. V3 is that there is an element 0 in our V, called the 0 vector, such that x + 0 = x, and also equals 0 + x. This is our additive identity.
4. Next is V4, which states that for any x in our vector space V, there exists an element –x such that x + (–x) = 0. This is called the additive inverse property.
5. V5 states that x + y = y + x, or in words, that addition is commutative.
6. V6 states that sx is an element of V, or in words, that our set V is closed under scalar multiplication.
7. V7 states that s(tx) should equal (st)x, which means that scalar multiplication is associative.
8. V8 states that the quantity (s + t) times the vector x should equal sx + tx. This says that scalar addition is distributive.
9. V9 states that s times the quantity (x + y) should equal sx + sy, which means that scalar multiplication is distributive.
10. And lastly, V10 states that the number 1 times x should equal x, which is the scalar multiplicative identity.

So matrix spaces and polynomial spaces can be naturally extended to complex vector spaces by simply allowing the entries or coefficients to be complex numbers. But eventually, as we did with real vector spaces, we would come to find that all finite-dimensional complex vector spaces are isomorphic to some Cn, so in this course, we will focus our attention on the study of Cn.