Determine the dimensions of the following spaces a) The range of the matrix:\[\begin{bmatrix}1 & 0 & 0 \\1 & 1 & 0 \\0 & 0 & 0 \\\end{bmatrix}\]b) The range of the matrix:\[\begin{bmatrix}1 & 0 & 0 \\1 & 1 & 1 \\0 & 1 & 1 \\\end{bmatrix}\]c) The span of the vectors $(1, -1, 1)$, $(-1, 1, 1)$, $(1, 1, -1)$, $(1, 1, 1)$.d) The span of the vectors $(1, 2, 3)$, $(2, 4, 6)$, $(3, 6, 9)$.e) $ P_3 $, the space of polynomials of degree less than or equal to 3.

Answered: a) range 1 0 0 11 00 10 b) range 1 1 01…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

Determine the dimensions of the following spaces

a) The range of the matrix:

\[
\begin{bmatrix}
1 & 0 & 0 \\
1 & 1 & 0 \\
0 & 0 & 0 \\
\end{bmatrix}
\]

b) The range of the matrix:

\[
\begin{bmatrix}
1 & 0 & 0 \\
1 & 1 & 1 \\
0 & 1 & 1 \\
\end{bmatrix}
\]

c) The span of the vectors $(1, -1, 1)$, $(-1, 1, 1)$, $(1, 1, -1)$, $(1, 1, 1)$.

d) The span of the vectors $(1, 2, 3)$, $(2, 4, 6)$, $(3, 6, 9)$.

e) $ P_3 $, the space of polynomials of degree less than or equal to 3.

Expert Solution

Step 1: We give definition of dimension.

Note : '' As per guidelines we will solve first three subparts. If you want any specific subpart to be solved please specify that subpart or post only that subpart.

Given spaces are,

$open parentheses a close parentheses space space space space space space r a n g e open parentheses open square brackets table row 1 0 0 row 1 1 0 row 0 0 0 end table close square brackets close parentheses$

$open parentheses b close parentheses space space space space space space r a n g e open parentheses open square brackets table row 1 0 0 row 1 1 1 row 0 1 1 end table close square brackets close parentheses$

$left parenthesis c right parenthesis space space space s p a n open curly brackets left parenthesis 1 comma negative 1 comma 1 right parenthesis space comma space left parenthesis negative 1 comma 1 comma 1 right parenthesis space comma space left parenthesis 1 comma 1 comma negative 1 right parenthesis space comma space left parenthesis 1 comma 1 comma 1 right parenthesis close curly brackets$

(.) Dimension: Number of elements in basis is called dimension.

(.) Basis : A linearly independent set of vectors which spans the vector space.

(.) Basis of range space is the set of linearly independent columns.