trouble 703
Using the an interpretation of the selection of a matrix, define the selection of the matrix\
Solution.
You are watching: What is the range of a matrix
by definition, the variety $\calR(A)$ of the procession $A$ is offered by\<\calR(A)=\left \ \quad A\mathbfx=\mathbfb \text for part \mathbfx \in \R^4 \right \.\>Thus, a vector $\mathbfb=\beginbmatrix b_1 \\ b_2 \\ b_3 \endbmatrix$ in $\R^3$ is in the selection $\calR(A)$ if and only if the system $A\mathbfx=\mathbfb$ is consistent. So, let us discover the conditions on $\mathbfb$ so the the mechanism is consistent.
To execute this, we take into consideration the augmented matrix of the system and also reduce it together follows. \beginalign*\left<\beginarrayrrrr 2 & 4 & 1 & -5 & b_1\\ 1 &2 & 1 & -2 & b_2 \\ 1 & 2 & 0 & -3 &b_3 \endarray\right> \xrightarrowR_1 \leftrightarrow R_2 \left<\beginarrayrrrr 1 &2 & 1 & -2 & b_2 \\ 2 & 4 & 1 & -5 & b_1\\ 1 & 2 & 0 & -3 &b_3 \endarray\right> \xrightarrow
Note that if the $(3, 5)$-entry $-b_1+b_2+b_3$ is not zero, climate the mechanism $A\mathbfx=\mathbf0$ is inconsistent due to the fact that this implies $0=1$.On the other hand, if $-b_1+b_2+b_3=0$, then we check out that the mechanism is consistent.Hence, the vector $\mathbfb$ is in the selection $\calR(A)$ if and only if $-b_1+b_2+b_3=0$.
See more: Which Of The Following Is A Good Predictor Of Biodiversity In Terrestrial Ecosystems?
In summary, us have\<\calR(A)=\left\ \quad -b_1+b_2+b_3=0 \right \.\>
Spanning collection for the range
With a little bit extr computation, we can uncover the spanning collection for the variety as follows.Thus, $\mathbfb \in \calR(A)$ if and only if\beginalign*\mathbfb=\beginbmatrix b_1 \\ b_2 \\ b_3 \endbmatrix=\beginbmatrix b_2+b_3 \\ b_2 \\ b_3 \endbmatrix=b_2\beginbmatrix 1 \\ 1 \\ 0 \endbmatrix+b_3\beginbmatrix 1 \\ 0 \\ 1 \endbmatrix.\endalign*
In summary, us have\beginalign*\calR(A)&=\left\ \mathbfb \in \R^3 \quad \middle \\<6pt> &=\Span\left\ \beginbmatrix 1 \\ 1 \\ 0 \endbmatrix, \beginbmatrix 1 \\ 0 \\ 1 \endbmatrix \right \. \endalign* Hence, the spanning set is \< \left\ \beginbmatrix 1 \\ 1 \\ 0 \endbmatrix, \beginbmatrix 1 \\ 0 \\ 1 \endbmatrix \right \.\>This spanning set is linearly independent, therefore it’s a basis for the range.