Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
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See our solution for Question 13E from Chapter 2.1 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 13E
Chapter:
Problem:
Let r1,…, rp be vectors in ℝn, and let Q be an m × n matrix. Write the matrix [Qr1,…,Qrp ] as a product of two matrices (neither of which is an identity matrix).
Step-by-Step Solution
Given Information
We are given with some vectors $r_1, r_2 ..r_p$ in $R^n$. We are also given with a matrix Q having dimensions $m \times n$. We have to write the entity [$Qr_1 \ \ Qr_2 \ \ .. ...Qr_p$] as a product of two matrices.
Step-1:
If the matrix A has dimensions $p \times r$ and the matrix B has dimensions $r \times s$. with columns $b_1, b_2, b_s$. Then the product AB has dimensions $p \times s$ having columns $Ab_1, Ab_2, Ab_s$. \[ \begin{aligned} A B & = A \left[ b _ { 1 } b _ { 2 } \ldots \ldots . b _ { s } \right] \\ & = \left[ A b _ { 1 } \ \ A b _ { 2 } \ \ . . . .A b _ { s } \right] \end{aligned} \] Hence the entity [$Qr_1 \ \ Qr_2 \ \ .. ...Qr_p$] is product of matrices Q and the matrix R formed by columns {$r_1, r_2 ..r_p$}
\[ \left[ \begin{array} { l l } { Q \mathbf { r } _ { 1 } } & { \dots } & { Q \mathbf { r } _ { p } } \end{array} \right] = Q R \]
We are given with some vectors $r_1, r_2 ..r_p$ in $R^n$. We are also given with a matrix Q having dimensions $m \times n$. We have to write the entity [$Qr_1 \ \ Qr_2 \ \ .. ...Qr_p$] as a product of two matrices.
Step-1:
If the matrix A has dimensions $p \times r$ and the matrix B has dimensions $r \times s$. with columns $b_1, b_2, b_s$. Then the product AB has dimensions $p \times s$ having columns $Ab_1, Ab_2, Ab_s$. \[ \begin{aligned} A B & = A \left[ b _ { 1 } b _ { 2 } \ldots \ldots . b _ { s } \right] \\ & = \left[ A b _ { 1 } \ \ A b _ { 2 } \ \ . . . .A b _ { s } \right] \end{aligned} \] Hence the entity [$Qr_1 \ \ Qr_2 \ \ .. ...Qr_p$] is product of matrices Q and the matrix R formed by columns {$r_1, r_2 ..r_p$}
\[ \left[ \begin{array} { l l } { Q \mathbf { r } _ { 1 } } & { \dots } & { Q \mathbf { r } _ { p } } \end{array} \right] = Q R \]