Matrix Representation of a Linear Transformation of the Vector Space $R^2$ to $R^2$

More from my site

• Give a Formula For a Linear Transformation From $\R^2$ to $\R^3$ Let $\{\mathbf{v}_1, \mathbf{v}_2\}$ be a basis of the vector space $\R^2$, where \[\mathbf{v}_1=\begin{bmatrix} 1 \\ 1 \end{bmatrix} \text{ and } \mathbf{v}_2=\begin{bmatrix} 1 \\ -1 \end{bmatrix}.\] The action of a linear transformation $T:\R^2\to \R^3$ on the […]
• Determine linear transformation using matrix representation Let $T$ be the linear transformation from the $3$-dimensional vector space $\R^3$ to $\R^3$ itself satisfying the following relations. \begin{align*} T\left(\, \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \,\right) =\begin{bmatrix} 1 \\ 0 \\ 1 […]
• Give the Formula for a Linear Transformation from $\R^3$ to $\R^2$ Let $T: \R^3 \to \R^2$ be a linear transformation such that \[T(\mathbf{e}_1)=\begin{bmatrix} 1 \\ 4 \end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix} 2 \\ 5 \end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix} 3 \\ 6 […]
• Give a Formula for a Linear Transformation if the Values on Basis Vectors are Known Let $T: \R^2 \to \R^2$ be a linear transformation. Let \[ \mathbf{u}=\begin{bmatrix} 1 \\ 2 \end{bmatrix}, \mathbf{v}=\begin{bmatrix} 3 \\ 5 \end{bmatrix}\] be 2-dimensional vectors. Suppose that \begin{align*} T(\mathbf{u})&=T\left( \begin{bmatrix} 1 \\ […]
• A Matrix Representation of a Linear Transformation and Related Subspaces Let $T:\R^4 \to \R^3$ be a linear transformation defined by \[ T\left (\, \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} \,\right) = \begin{bmatrix} x_1+2x_2+3x_3-x_4 \\ 3x_1+5x_2+8x_3-2x_4 \\ x_1+x_2+2x_3 \end{bmatrix}.\] (a) Find a matrix $A$ such that […]
• Prove a Given Subset is a Subspace and Find a Basis and Dimension Let \[A=\begin{bmatrix} 4 & 1\\ 3& 2 \end{bmatrix}\] and consider the following subset $V$ of the 2-dimensional vector space $\R^2$. \[V=\{\mathbf{x}\in \R^2 \mid A\mathbf{x}=5\mathbf{x}\}.\] (a) Prove that the subset $V$ is a subspace of $\R^2$. (b) Find a basis for […]
• Matrix of Linear Transformation with respect to a Basis Consisting of Eigenvectors Let $T$ be the linear transformation from the vector space $\R^2$ to $\R^2$ itself given by \[T\left( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \right)= \begin{bmatrix} 3x_1+x_2 \\ x_1+3x_2 \end{bmatrix}.\] (a) Verify that the […]
• Vector Space of Polynomials and Coordinate Vectors Let $P_2$ be the vector space of all polynomials of degree two or less. Consider the subset in $P_2$ \[Q=\{ p_1(x), p_2(x), p_3(x), p_4(x)\},\] where \begin{align*} &p_1(x)=x^2+2x+1, &p_2(x)=2x^2+3x+1, \\ &p_3(x)=2x^2, &p_4(x)=2x^2+x+1. \end{align*} (a) Use the basis […]