I am having a really hard time trying to understand what "the image of A" means and how to start such a question. I was thinking of creating a matrix with b1 as the augumented part of the matrix.
determine whether the vector is in the image of A
0
$\begingroup$
matrices
2 Answers
1
The image of a function $f: X \rightarrow Y$ is the set of elements of $Y$ that are the result of applying $f$ to some $x$ in $X$.
Any $3\times 3$ matrix $A$ gives rise to a linear transformation $T_A$ given by $T_A(\vec{x})=A \vec{x}$.
In other words, a vector $\vec{b}$ is in the image of $A$ if there exists a vector $\vec{x}$ such that $A\vec{x} = \vec{b}$.
Hint: What has this got to do with solving a system of linear equations?
0
The hint is in the text of your exercise. The image of $A$ is the abbreviation for the image of a linear map defined by the matrix $A$, i.e. $T:\Bbb R^3\to\Bbb R^3$ given by $T(x)=Ax$. To check whether a given vector $y$ lies in this image, you need to find the vector $x$ s.t. $y=Ax$. This leads to the system of linear equations.