91Ó°ÊÓ

TRUE OR FALSE

42. If “If $x_1,x_2,...,x_n$are any real numbers then the inequality

${\left(\underset{k=1}{\overset{n}{∑}}{x}_i\right)}^2≤n\underset{k=1}{\overset{n}{∑}}\left(x_k^2\right)$

must hold.

Consider a unit vector$\stackrel{\mathbf{âŠ\yen }}{\mathbf{u}}$in ${\mathit{R}}^{\mathbf{3}}$. We define the matrices

$\mathit{A}\mathbf{=}\mathbf{2}\stackrel{\mathbf{âŠ\yen }}{\mathbf{u}}\stackrel{\mathbf{âŠ\yen }}{\mathbf{u}}\mathbf{}\mathbf{-}{\mathit{l}}_{\mathbf{3}}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{B}\mathbf{=}{\mathit{l}}_{\mathbf{3}}\mathbf{-}\mathbf{2}\stackrel{\mathbf{âŠ\yen }}{\mathbf{u}}\stackrel{\mathbf{âŠ\yen }}{\mathbf{u}}\mathbf{}$

Describe the linear transformations defined by these matrices geometrically

Let Abe the matrix of an orthogonal projection. Find${\mathit{A}}^{\mathbf{2}}$ in two ways:

a.Geometrically. (Consider what happens when you apply an orthogonal projection twice.)

b.By computation, using the formula given in Theorem 5.3.10

Consider a block matrix $\mathit{A}\mathbf{=}\left[\begin{array}{cc}{A}_1,& A_2\end{array}\right]$ with linearly independent columns. (A1 is an $\mathit{n}\mathbf{×}{\mathit{m}}_{\mathbf{1}}$ matrix, and A2 is $\mathit{n}\mathbf{×}{\mathit{m}}_{\mathbf{2}}$.) Suppose you know the QR factorization of A. Explain how this allows you to find the QRfactorization of ${\mathit{A}}_{\mathbf{1}}$.

In Exercises 40 through 46, consider vectors$\stackrel{\mathbf{→}}{{\mathbf{V}}_{\mathbf{1}}}\mathbf{,}\stackrel{\mathbf{→}}{{\mathbf{V}}_{\mathbf{2}}}\mathbf{,}\stackrel{\mathbf{→}}{{\mathbf{V}}_{\mathbf{3}}}$in${\mathbf{R}}^{\mathbf{4}}$; we are told that$\stackrel{\mathbf{→}}{{\mathbf{V}}_{\mathbf{i}}}\mathbf{,}\stackrel{\mathbf{→}}{{\mathbf{V}}_{\mathbf{j}}}$is the entry ${\mathbf{a}}_{\mathbf{ij}}$of matrix A.

$\mathbf{A}\mathbf{=}\left[\begin{array}{ccc}3& 5& 11\\ 5& 9& 20\\ 11& 20& 49\end{array}\right]$

Find ${\mathbf{proj}}_{\stackrel{\mathbf{→}}{\mathbf{v}}}\left({\stackrel{→}{v}}_1\right)$, expressed as a scalar multiple of${\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{2}}$.

Determine whether the statement “If $\mathit{A}{\mathit{A}}^{\mathbf{T}}\mathbf{=}{\mathit{A}}^{\mathbf{2}}$for$\mathbf{2}\mathbf{×}\mathbf{2}$ matrix A then A must be symmetric.

Consider an n×mmatrix Awith rank(A) <m. Is it always possible to write A=QRWhere Qis an n×mmatrix with orthonormal columns and Ris upper triangular? Explain.

In Exercises 40 through 46, consider vectors $\mathit{v}{\mathbf{⃗}}_{\mathbf{1}}\mathbf{,}\mathit{v}{\mathbf{⃗}}_{\mathbf{2}}\mathbf{,}\mathit{v}{\mathbf{⃗}}_{\mathbf{3}}$in ${\mathit{R}}^{\mathbf{4}}$; we are told that $\mathit{v}{\mathbf{⃗}}_{\mathbf{i}}\mathbf{,}\mathit{v}{\mathbf{⃗}}_{\mathbf{j}}$is the entry ${\mathit{a}}_{\mathbf{i}\mathbf{j}}$of matrix A.

$\mathit{A}\mathbf{=}\left[\begin{array}{ccc}3& 5& 11\\ 5& 9& 20\\ 11& 20& 49\end{array}\right]$

Find a nonzero vector $\stackrel{\mathbf{â‡\P Ä}}{\mathbf{v}}$in span $\left({\stackrel{â‡\P Ä}{v}}_2{\stackrel{â‡\P Ä}{v}}_3\right)$such that $\stackrel{\mathbf{â‡\P Ä}}{\mathbf{v}}$is orthogonal to ${\stackrel{\mathbf{â‡\P Ä}}{\mathbf{v}}}_{\mathbf{3}}$.Express as a linear combination of localid="1659441496004" ${\stackrel{\mathbf{â‡\P Ä}}{\mathbf{v}}}_{\mathbf{2}}$and ${\stackrel{\mathbf{â‡\P Ä}}{\mathbf{v}}}_{\mathbf{3}}$.

TRUE OR FALSE

If V is a subspace of ${\mathit{R}}^{\mathbf{n}}$and $\stackrel{\mathbf{→}}{\mathbf{x}}$is a vector space in ${\mathit{R}}^{\mathbf{n}}$, then the inequality $\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{·}\mathbf{\left(}\mathbf{pr}{\mathbf{o}}_{\mathbf{V}}\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{\right)}\mathbf{â\copyright ¾}\mathbf{0}$must hold.

For which n×mmatrices Adoes the equation

$\mathit{d}\mathit{i}\mathit{m}\left(ker\left(A\right)\right)\mathbf{=}\mathit{d}\mathit{i}\mathit{m}\left(ker\left(A^T\right)\right)$hold? Explain.