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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q15E (page 246)

Consider an $m 脳 n$ matrix A with $K e r (A) = {\overset{鈫�}{0}}$ . Show that there exists an $n 脳 m$ matrix B such that $B A = l_{n}$ .

Short Answer

Expert verified

The matrix is ${(A^{T} A)}^{- 1} - A^{T}$ .

Step by step solution

01

Determine the matrix R.

Consider a $m 脳 n$ matrix A.

If $K e r (A) = {\overset{鈫�}{0}}$ for arole="math" localid="1659500428667" $m 脳 n$ matrix A then the matrix A is invertible.

If the matrix A is invertible then the matrixrole="math" localid="1659500442729" $A^{T}$ is invertible.

If the matrices A and B is invertible then the matrix AB is invertible.

As the value of $Ke r (A) = {\overset{鈫�}{0}}$ , by the definition the matrix A is invertible.

By the definition, the matrices $A^{T}$ and $A^{T} A$ are invertible.

Assume the matrix $B A = {(A^{T} A)}^{- 1}$ , simplify the matrix BA as follows.

$\begin{array}{l} B A = A^{- 1} \{l_{n}\} A \\ = A^{- 1} {(A^{T})}^{- 1} A^{T} A \\ = A^{- 1} \{{(A^{T})}^{- 1} A^{T}\} A \\ B A = A^{- 1} (l_{n}) A \end{array}$

Further, simplify the equation as follows.

$\begin{array}{l} B A = A^{- 1} \{l_{n}\} A \\ = A^{- 1} A \\ B A = l_{n} \end{array}$

Hence, for the matrix $B = {(A^{T} A)}^{- 1} - A^{T}$ the equation $B A = l_{n}$ .

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Most popular questions from this chapter

TRUE OR FALSE? The equation $d e t (A^{T}) = d e t (A)$ holds for all $2 脳 2$ matrices A.

Find the angle $胃$ between each of the pairs of vectors $\overset{鈬赌}{u}$ and $\overset{鈬赌}{v}$ in exercises 4 through 6.

4. $\overset{鈬赌}{u} = [\begin{matrix} 1 \\ 1 \end{matrix}], \overset{鈬赌}{v} = [\begin{matrix} 7 \\ 11 \end{matrix}] .$

In Exercises 40 through 46, consider vectors in ${\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, {\overset{鈫�}{v}}_{3}$ ; we are told thatrole="math" localid="1659495854834" ${\overset{鈫�}{v}}_{i}, {\overset{鈫�}{v}}_{j}$ is the entry $a_{i j}$ of matrix A.

$A = [\begin{matrix} 3 & 5 & 11 \\ 5 & 9 & 20 \\ 11 & 20 & 49 \end{matrix}]$

46. Find $p r o j_{v} ({\overset{鈫�}{v}}_{3})$ , where V =span role="math" localid="1659495997207" $({\overset{鈫�}{v}}_{1} . {\overset{鈫�}{v}}_{2})$ . Express your answer as a linear combination ofrole="math" localid="1659496026018" ${\overset{鈫�}{v}}_{1}$ and ${\overset{鈫�}{v}}_{2}$ .

Find the length of each of the vectors $\overset{鈬赌}{v}$ In exercises 1 through 3.

$\overset{鈬赌}{v} = [\begin{matrix} 7 \\ 11 \end{matrix}]$

Consider the orthonormal vectors ${\overset{鈫�}{u}}_{1}, {\overset{鈫�}{u}}_{2}, {\overset{鈫�}{u}}_{3}, {\overset{鈫�}{u}}_{4}, {\overset{鈫�}{u}}_{5}$ in ${鈩�}^{10}$ . Find the length of the vector $\overset{鈫�}{x} = 7 {\overset{鈫�}{u}}_{1} - 3 {\overset{鈫�}{u}}_{2} + 2 {\overset{鈫�}{u}}_{3} + {\overset{鈫�}{u}}_{4} - {\overset{鈫�}{u}}_{5}$ .

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