As in Problem 72, let V be the set of all vectors x⃗ in ℝ^4 such that x3=x1+x2 and x4=x2+x3 . In

step 1 For this exercise, we have x, uh, t of x and x equal to x sub 4, negative x of 3, x of 2, negati

As in Problem 72, let V be the set of all vectors x⃗ in ℝ^4...

As in Problem $72,$ let $V$ be the set of all vectors $\vec{x}$ in $\mathbb{R}^{4}$ such that $x_{3}=x_{1}+x_{2}$ and $x_{4}=x_{2}+x_{3} .$ In Problem 72 we see that $V$ is a subspace of $\mathbb{R}^{4}$ with $\operatorname{dim}(V)=2$ a. Consider the linear transformation $T\left[\begin{array}{l}x_{1} \\ x_{2} \\ x_{3} \\ x_{4}\end{array}\right]=$ $\left[\begin{array}{r}x_{4} \\ -x_{3} \\ x_{2} \\ -x_{1}\end{array}\right]$ from $\mathbb{R}^{4}$ to $\mathbb{R}^{4} .$ Show that $T(\vec{x})$ is orthogonal to $\vec{x},$ for all $\vec{x}$ in $\mathbb{R}^{4} .$ If $\vec{x}$ is a vector in $V,$ show that $T(\vec{x})$ is in $V$ as well. Thus, $T$ induces a linear transformation from $V$ to $V,$ which we will denote by $F$. b. Find the matrix $A$ of $F$ with respect to the basis $$\mathfrak{A}=\left(\left[\begin{array}{l} 1 \\ 0 \\ 1 \\ 1 \end{array}\right],\left[\begin{array}{l} 0 \\ 1 \\ 1 \\ 2 \end{array}\right]\right)$$ c. Find the matrix $B$ of $F$ with respect to the basis $$\mathfrak{B}=\left(\left[\begin{array}{l} 0 \\ 1 \\ 1 \\ 2 \end{array}\right],\left[\begin{array}{r} 2 \\ -1 \\ 1 \\ 0 \end{array}\right]\right)$$ d. Find the change of basis matrix $S=S_{\mathfrak{B} \rightarrow \mathfrak{A}}$ e. Write an equation relating the matrices $A, B,$ and $S$ and check that this equation holds for the matrices you found in parts (b), (c), and (d). f. Does there exist a basis $\mathbb{G}$ of $V$ such that the $\mathbb{G}$ matrix $C$ of $F$ is diagonal?

As in Problem $72,$ let $V$ be the set of all vectors $\vec{x}$ in $\mathbb{R}^{4}$ such that $x_{3}=x_{1}+x_{2}$ and $x_{4}=x_{2}+x_{3} .$ In Problem 72 we see that $V$ is a subspace of $\mathbb{R}^{4}$ with $\operatorname{dim}(V)=2$
a. Consider the linear transformation $T\left[\begin{array}{l}x_{1} \\ x_{2} \\ x_{3} \\ x_{4}\end{array}\right]=$ $\left[\begin{array}{r}x_{4} \\ -x_{3} \\ x_{2} \\ -x_{1}\end{array}\right]$ from $\mathbb{R}^{4}$ to $\mathbb{R}^{4} .$ Show that $T(\vec{x})$ is orthogonal to $\vec{x},$ for all $\vec{x}$ in $\mathbb{R}^{4} .$ If $\vec{x}$ is a vector in $V,$ show that $T(\vec{x})$ is in $V$ as well. Thus, $T$ induces a linear transformation from $V$ to $V,$ which we will denote by $F$.
b. Find the matrix $A$ of $F$ with respect to the basis
$$\mathfrak{A}=\left(\left[\begin{array}{l}
1 \\
0 \\
1 \\
1
\end{array}\right],\left[\begin{array}{l}
0 \\
1 \\
1 \\
2
\end{array}\right]\right)$$
c. Find the matrix $B$ of $F$ with respect to the basis
$$\mathfrak{B}=\left(\left[\begin{array}{l}
0 \\
1 \\
1 \\
2
\end{array}\right],\left[\begin{array}{r}
2 \\
-1 \\
1 \\
0
\end{array}\right]\right)$$
d. Find the change of basis matrix $S=S_{\mathfrak{B} \rightarrow \mathfrak{A}}$
e. Write an equation relating the matrices $A, B,$ and $S$ and check that this equation holds for the matrices you found in parts (b), (c), and (d).
f. Does there exist a basis $\mathbb{G}$ of $V$ such that the $\mathbb{G}$ matrix $C$ of $F$ is diagonal?

Key Concepts

Diagonalization

Diagonalization is the process of finding a basis in which the matrix representation of a linear transformation is diagonal. This essentially simplifies the transformation to scaling along the directions of the basis vectors and is closely associated with the existence of eigenvalues and eigenvectors.

Orthogonality

Orthogonality refers to a relationship between vectors where their inner product (or dot product in Euclidean spaces) is zero, indicating that they are perpendicular. This concept is important in many areas of linear algebra, including the examination of properties of transformations and subspaces.

Matrix Representation of a Linear Transformation

A linear transformation can be represented as a matrix once a basis for the domain and codomain is chosen. This matrix encapsulates the transformation rules and enables the computation of images of vectors and the analysis of the transformation’s properties using linear algebra techniques.

Change of Basis

The change of basis is the process of converting the coordinate representation of vectors or the matrix of a linear transformation from one basis to another. This involves using a transition matrix that relates the bases and often simplifies the understanding or computation within the vector space.

Dimension

The dimension of a subspace refers to the number of linearly independent vectors in any basis for that subspace. This number indicates the minimal amount of information needed to describe every vector in the subspace through linear combinations.

Vector Subspace

A vector subspace is a subset of a larger vector space that is closed under addition and scalar multiplication. It inherits all the properties of a vector space and has a well?defined dimension that corresponds to the number of basis vectors required to span it.

Linear Transformation

A linear transformation is a function between vector spaces that preserves the operations of vector addition and scalar multiplication. It can be completely characterized by its action on a basis of the domain, and it allows abstract operations to be represented concretely via matrices.

Basis

A basis is a set of linearly independent vectors that spans a vector space or subspace. It provides the coordinate system relative to which vectors in the space can be uniquely represented by their coordinates.

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