Find the change-of-basis matrices and the matrix representations of a linear transformation in the standard basis and a new basis.

Matrix Representation and Change of Basis in R^2

Linear transformations core

Consider the linear transformation $F$ on $\mathbb{R}^2$ defined by $F(x,y) = (5x - y, 2x + y)$ and the following bases of $\mathbb{R}^2$: \[ E = \{e_1, e_2\} = \{(1,0), (0,1)\} \] and \[ S = \{u_1, u_2\} = \{(1,4), (2,7)\} \] (a) Find the change-of-basis matrix $P$ from $E$ to $S$ and the change-of-basis matrix $Q$ from $S$ back to $E$. (b) Find the matrix $A$ that represents $F$ in the basis $E$. (c) Find the matrix $B$ that represents $F$ in the basis $S$.

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What you need first

Matrix multiplication
Finding the inverse of a 2x2 matrix
Solving systems of linear equations
Understanding coordinates relative to a basis

What this problem tests

Tests the ability to compute change-of-basis matrices, find coordinates relative to a new basis, and apply the change-of-basis formula for linear transformations.

This is a standard problem in a first course in linear algebra, testing the understanding of how linear transformations are represented by matrices and how these representations change when the basis is changed.

Matrix Representation and Change of Basis in R^2

What you need first

What this problem tests

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