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# Prove that the unit coordinate vectors form a basis for Cn

The unit coordinate vectors in are defined by

where the 1 is in the th position. We know that these form a basis for . Prove that they also form a basis for .

Proof. First, span since if is any vector then

Further, are independent in since

Hence, . Therefore, the unit coordinate vectors form a basis for

# Prove that a given set of vectors forms a basis for C3

1. Let

be three vectors in . Prove that these vectors form a basis for .

2. Write the vector as a linear combination of .

1. Proof. We know that any set of three linearly independent vectors in will span , and thus form a basis. (This is from Theorem 12.10, which is valid for .) Thus, it is sufficient to show that are linearly independent. To that end, let be scalars in , then

From the third equation we have , and so the second equation implies , and finally the third equation implies . Hence, , and are linearly independent

2. To express as a linear combination of , let be scalars. Then,

From the third equation we have . Plugging this into the first and second equations we get and . Therefore,

# Compute the angle between (1,0,i,i,i) and (i,i,i,0,i) in C5

We define the angle between two vectors and in by the formula

Compute the angle between and in .

We compute as follows:

# Prove an identity for the angle between vectors in Cn

The angle between two vectors non-zero is defined by the equation

The inequality

we established in the previous exercise (Section 12.17, Exercise #6) show that there is a unique satisfying this equation. Prove that we have

Proof. From the definition of we have

But then we know from this exercise (Section 12.17, Exercise #3) that

And, we know from this exercise (Section 12.17, Exercise #5) that

Therefore,

# Prove some facts about vectors in Cn

1. Prove that for we have

2. For non-zero vectors prove that

1. Proof. Let for .

2. Proof. Let for . Then,

and

Therefore,

Therefore,

# Prove yet another identity for vectors in Cn

Let be any two vectors. Prove that we have the following identity,

Proof. Using our computations of and in the previous exercise (Section 12.17, Exercise #4) we have

# Prove another identity for vectors in Cn

Let be any two vectors. Prove that we have the following identity:

Proof. From the previous exercise (Section 12.17, #3) we have the identity

This also give us

Therefore,

# Prove an identity for vectors in Cn

Let be any two vectors. Prove the following identity:

Proof. We can compute, noting that by Theorem 12.11(a),

# Find a nonzero vector in C3 orthogonal to two given vectors

Let

Find a non-zero vector in orthogonal to both and .

For to be orthogonal to both and we must have and . This gives us

From the first equation we get . Plugging this into the second equation we then have . Let , then we have

Therefore, is a vector in orthogonal to both and .

# Compute some dot products of vectors in C2

Let

be vectors in . Compute the following dot products.

1. ;
2. ;
3. ;
4. ;
5. ;
6. ;
7. ;
8. ;
9. ;
10. .

1. We compute,

2. We compute,

3. We compute,

4. We compute,

5. We compute,

6. We compute,

7. We compute,

8. We compute,

9. We compute,

10. We compute,