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I'm new to the concept, so solve it and explain it to me with fundamental concepets need to know to solve this question. 

### Problem Breakdown

The problem asks you to find the **representation of a vector** \( \mathbf{x} = \begin{bmatrix} 1 \\ 3 \end{bmatrix} \) in two different bases:

1. **Standard basis** \( [\mathbf{e_1}, \mathbf{e_2}] \), where:
   - \( \mathbf{e_1} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \)
   - \( \mathbf{e_2} = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \)
   
2. **Another basis** \( [\mathbf{\tilde{e}_1}, \mathbf{\tilde{e}_2}] \), where:
   - \( \mathbf{\tilde{e}_1} = \begin{bmatrix} 1 \\ 4 \end{bmatrix} \)
   - \( \mathbf{\tilde{e}_2} = \begin{bmatrix} 3 \\ 2 \end{bmatrix} \)

The aim is to express the vector \( \mathbf{x} \) in terms of these two bases.

### Key Concepts Needed

1. **Vector Representation in a Basis:**
   - A vector can be represented as a linear combination of basis vectors. For example, if \( \mathbf{v} \) is a vector, then we can write:
     \[
     \mathbf{v} = a_1 \mathbf{b_1} + a_2 \mathbf{b_2}
     \]
     Here, \( a_1 \) and \( a_2 \) are the coordinates of \( \mathbf

Consider the two-dimensional vector space (R^2, R) with vectors x = [1 3]^T, e1 = [1 0]^T, e2 = [0 1]^T, e_1 = [1 4]^T, e_2 = [3 2]^T. Find the representation of x with respect to [e1, e2] and [e_1, e_2].

Learn how to find the representation of a vector in two different bases for a 2D vector space. This problem covers key linear algebra concepts, such as basis vectors and linear combinations. Suitable for university-level students studying linear algebra.

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