$\mathbf{Q 4}\left[5\right.$ marks] Let $V=(V,+, \cdot, \mathbb{R})$ be a vector space and let $B=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\} \subseteq V$ be a basis.
a) Define what it means that the triple $\left(a_{1}, a_{2}, a_{3}\right) \in \mathbb{R}^{3}$ is the coordinate vector of $\mathbf{x} \in V$ with respect to the basis $B$.
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School of Mathematics and Statistics
University of New South Wales
b) Find the coordinate vector of $\mathbf{v}_{1}$ with respect to the basis $B$.
c) Define what it means that a matrix $A \in M_{3,3}(\mathbb{R})$ is the matrix of the linear map $T: V \rightarrow V$ with respect to the basis $B$.
d) Let
$$
A=\left(\begin{array}{lll}
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 0
\end{array}\right) .
$$
be the matrix of the map $T$ with respect to the basis $B$. Find $T\left(\mathbf{v}_{1}\right)$ in terms of $B$.

Question

$\mathbf{Q 4}\left[5\right.$ marks] Let $V=(V,+, \cdot, \mathbb{R})$ be a vector space and let $B=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\} \subseteq V$ be a basis.
a) Define what it means that the triple $\left(a_{1}, a_{2}, a_{3}\right) \in \mathbb{R}^{3}$ is the coordinate vector of $\mathbf{x} \in V$ with respect to the basis $B$.
Page 1 of 2

School of Mathematics and Statistics
University of New South Wales
b) Find the coordinate vector of $\mathbf{v}_{1}$ with respect to the basis $B$.
c) Define what it means that a matrix $A \in M_{3,3}(\mathbb{R})$ is the matrix of the linear map $T: V \rightarrow V$ with respect to the basis $B$.
d) Let
$$
A=\left(\begin{array}{lll}
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 0
\end{array}\right) .
$$
be the matrix of the map $T$ with respect to the basis $B$. Find $T\left(\mathbf{v}_{1}\right)$ in terms of $B$.

Accepted Answer

∻Solution by Steps∻ ### Part (a) ‖ step 1 ⋮ The coordinate vector of $\mathbf{x} \in V$ with respect to the basis $B = \{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is a unique ordered triple $(a_1, a_2, a_3) \in \mathbb{R}^3$ such that ‖ ‖ step 2 ⋮ $\mathbf{x} = a_1 \mathbf{v}_1 + a_2 \mathbf{v}_2 + a_3 \mathbf{v}_3$ ‖ ### Part (b) ‖ step 1 ⋮ The coordinate vector of $\mathbf{v}_1$ with respect to the basis $B$ is found by expressing $\mathbf{v}_1$ as a linear combination of the basis vectors. ‖ ‖ step 2 ⋮ Since $\mathbf{v}_1$ is itself a basis vector, its coordinate vector is $(1, 0, 0)$. ‖ ### Part (c) ‖ step 1 ⋮ A matrix $A \in M_{3,3}(\mathbb{R})$ is the matrix of the linear map $T: V ightarrow V$ with respect to the basis $B$ if ‖ ‖ step 2 ⋮ $A$ is the matrix whose columns are the coordinate vectors of $T(\mathbf{v}_1), T(\mathbf{v}_2), T(\mathbf{v}_3)$ with respect to $B$. ‖ ### Part (d) ‖ step 1 ⋮ Given the matrix $A = \begin{pmatrix} 0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 0 \end{pmatrix}$, we need to find $T(\mathbf{v}_1)$ in terms of $B$. ‖ ‖ step 2 ⋮ The first column of $A$ represents the coordinate vector of $T(\mathbf{v}_1)$ with respect to $B$. ‖ ‖ step 3 ⋮ Therefore, $T(\mathbf{v}_1) = 0 \mathbf{v}_1 + 1 \mathbf{v}_2 + 0 \mathbf{v}_3 = \mathbf{v}_2$. ‖ ∻Answer∻ ⚹ $T(\mathbf{v}_1) = \mathbf{v}_2$ ⚹ ∻Key Concept∻ ⚹Coordinate Vector⚹ ∻Explanation∻ ⚹The coordinate vector of a vector $\mathbf{x}$ with respect to a basis $B$ is the unique ordered triple that expresses $\mathbf{x}$ as a linear combination of the basis vectors.⚹◊What is the definition of the coordinate vector of a vector with respect to a basis in a vector space⍭ Generate me a similar question◊

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