Find an Orthonormal Basis of the Range of a Linear Transformation

Problem 478

Let $T:\R^2 \to \R^3$ be a linear transformation given by
\[T\left(\, \begin{bmatrix}
x_1 \\
x_2
\end{bmatrix} \,\right)
=
\begin{bmatrix}
x_1-x_2 \\
x_2 \\
x_1+ x_2
\end{bmatrix}.\] Find an orthonormal basis of the range of $T$.

(The Ohio State University, Linear Algebra Final Exam Problem)

 
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A Linear Transformation Preserves Exactly Two Lines If and Only If There are Two Real Non-Zero Eigenvalues

Problem 472

Let $T:\R^2 \to \R^2$ be a linear transformation and let $A$ be the matrix representation of $T$ with respect to the standard basis of $\R^2$.

Prove that the following two statements are equivalent.

(a) There are exactly two distinct lines $L_1, L_2$ in $\R^2$ passing through the origin that are mapped onto themselves:
\[T(L_1)=L_1 \text{ and } T(L_2)=L_2.\]

(b) The matrix $A$ has two distinct nonzero real eigenvalues.

 
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Use the Cayley-Hamilton Theorem to Compute the Power $A^{100}$

Problem 471

Let $A$ be a $3\times 3$ real orthogonal matrix with $\det(A)=1$.

(a) If $\frac{-1+\sqrt{3}i}{2}$ is one of the eigenvalues of $A$, then find the all the eigenvalues of $A$.

(b) Let
\[A^{100}=aA^2+bA+cI,\] where $I$ is the $3\times 3$ identity matrix.
Using the Cayley-Hamilton theorem, determine $a, b, c$.

(Kyushu University, Linear Algebra Exam Problem)
 
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Diagonalize a 2 by 2 Matrix $A$ and Calculate the Power $A^{100}$

Problem 466

Let
\[A=\begin{bmatrix}
1 & 2\\
4& 3
\end{bmatrix}.\]

(a) Find eigenvalues of the matrix $A$.

(b) Find eigenvectors for each eigenvalue of $A$.

(c) Diagonalize the matrix $A$. That is, find an invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

(d) Diagonalize the matrix $A^3-5A^2+3A+I$, where $I$ is the $2\times 2$ identity matrix.

(e) Calculate $A^{100}$. (You do not have to compute $5^{100}$.)

(f) Calculate
\[(A^3-5A^2+3A+I)^{100}.\] Let $w=2^{100}$. Express the solution in terms of $w$.

 
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Are Linear Transformations of Derivatives and Integrations Linearly Independent?

Problem 463

Let $W=C^{\infty}(\R)$ be the vector space of all $C^{\infty}$ real-valued functions (smooth function, differentiable for all degrees of differentiation).
Let $V$ be the vector space of all linear transformations from $W$ to $W$.
The addition and the scalar multiplication of $V$ are given by those of linear transformations.

Let $T_1, T_2, T_3$ be the elements in $V$ defined by
\begin{align*}
T_1\left(\, f(x) \,\right)&=\frac{\mathrm{d}}{\mathrm{d}x}f(x)\\[6pt] T_2\left(\, f(x) \,\right)&=\frac{\mathrm{d}^2}{\mathrm{d}x^2}f(x)\\[6pt] T_3\left(\, f(x) \,\right)&=\int_{0}^x \! f(t)\,\mathrm{d}t.
\end{align*}
Then determine whether the set $\{T_1, T_2, T_3\}$ are linearly independent or linearly dependent.

 
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