The answer: 1) 𝑒1
⃗ ⃗ ⃗⃗
= ( 3
−5) , 𝑒2
⃗ ⃗ ⃗⃗
= (4
0) The question: Find the basic vectors of the basis E

⃗⃗

,

If:

1) a

⃗

= (22

−10), b

⃗⃗

= (21

5 ) and

A
⃗
E
⃗ ⃗ = (2 4), b
⃗⃗
E
⃗ ⃗ = (-1 6 ) coordinate vectors in the basis E

⃗⃗

.

To find the basic vectors of the basis \( E \) from the information given, we’ll proceed as follows:

### Step 1: Analyze Given Basis Vectors
We are given:
\[
\mathbf{e}_1 = \begin{pmatrix} 3 \\ -5 \end{pmatrix}, \quad \mathbf{e}_2 = \begin{pmatrix} 4 \\ 0 \end{pmatrix}
\]
These vectors form the basis \( E \).

### Step 2: Write Down Coordinate Transformations
The question also provides coordinates for two vectors \( \mathbf{a} \) and \( \mathbf{b} \) in terms of both the standard basis and basis \( E \):
1. Vector \( \mathbf{a} \) in the standard basis is \( \mathbf{a} = \begin{pmatrix} 22 \\ -10 \end{pmatrix} \).
2. The coordinates of \( \mathbf{a} \) in basis \( E \) are given as \( \mathbf{a}_E = \begin{pmatrix} 2 \\ 4 \end{pmatrix} \).
3. Vector \( \mathbf{b} \) in the standard basis is \( \mathbf{b} = \begin{pmatrix} 21 \\ 5 \end{pmatrix} \).
4. The coordinates of \( \mathbf{b} \) in basis \( E \) are given as \( \mathbf{b}_E = \begin{pmatrix} -1 \\ 6 \end{pmatrix} \).

### Step 3: Verify Basis Vectors of \( E \)
To confirm that \(\mathbf{e}_1\) and \(\mathbf{e}_2\) form the basis \( E \), we can check if \( \mathbf{a} \) and \( \mathbf{b} \) can be expressed as linear combinations of \(\mathbf{e}_1\) and \(\mathbf{e}_2\) using the provided coordinates in basis \( E \).

#### For \( \mathbf{a} \):
\[
\mathbf{a} = 2 \cdot \mathbf{e}_1 + 4 \cdot \mathbf{e}_2 = 2 \begin{pmatrix} 3 \\ -5 \end{pmatrix} + 4 \begin{pmatrix} 4 \\ 0 \end{pmatrix}
\]
Calculating this:
\[
\mathbf{a} = \begin{pmatrix} 6 \\ -10 \end{pmatrix} + \begin{pmatrix} 16 \\ 0 \end{pmatrix} = \begin{pmatrix} 22 \\ -10 \end{pmatrix}
\]
This matches the given \( \mathbf{a} \).

#### For \( \mathbf{b} \):
\[
\mathbf{b} = -1 \cdot \mathbf{e}_1 + 6 \cdot \mathbf{e}_2 = -1 \begin{pmatrix} 3 \\ -5 \end{pmatrix} + 6 \begin{pmatrix} 4 \\ 0 \end{pmatrix}
\]
Calculating this:
\[
\mathbf{b} = \begin{pmatrix} -3 \\ 5 \end{pmatrix} + \begin{pmatrix} 24 \\ 0 \end{pmatrix} = \begin{pmatrix} 21 \\ 5 \end{pmatrix}
\]
This matches the given \( \mathbf{b} \).

### Conclusion
The basis \( E \) is confirmed to be:
\[
\mathbf{e}_1 = \begin{pmatrix} 3 \\ -5 \end{pmatrix}, \quad \mathbf{e}_2 = \begin{pmatrix} 4 \\ 0 \end{pmatrix}
\]

Let me know if you'd like additional details or have questions. 

---

**Further Questions:**
1. How can we find the inverse of a coordinate transformation?
2. What are the properties of a vector space basis?
3. How can we check if a set of vectors forms a basis?
4. What is the significance of linear independence in a basis?
5. How do coordinate transformations affect vector norms?

**Tip:** Always verify that your calculated coordinates match the given ones when dealing with different bases to ensure accuracy.

The answer: e1 = (3, -5), e2 = (4, 0). The question: Find the basic vectors of the basis E. If: a = (22, -10), b = (21, 5) and AE = (2, 4), bE = (-1, 6) as coordinate vectors in the basis E.

Explore how to determine basis vectors and perform coordinate transformations in linear algebra. This problem includes key concepts such as linear independence, basis definitions, and vector representation in different bases. Suitable for college-level or advanced high school students.

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