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.

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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.