Note that if \(\left[\begin{array}{c} a_1 \\ \vdots \\ a_n \end{array}\right] \) and \(\left[\begin{array}{c} b_1 \\ \vdots \\ b_n \end{array}\right] \) are both solutions, we know that
shows that \(\left[\begin{array}{c} a_1+ b_1 \\ \vdots \\ a_n+b_n \end{array}\right] \) is also a solution. Thus the solution set of a homogeneous system is...
Closed under addition.
Not closed under addition.
Linearly dependent.
Linearly independent.
(b)
Similarly, if \(c \in \IR\text{,}\)\(\left[\begin{array}{c} ca_1 \\ \vdots \\ ca_n \end{array}\right] \) is a solution. Thus the solution set of a homogeneous system is also closed under scalar multiplication, and therefore...
Which of these choices best describes the set of two vectors \(\left\{\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\end{array}\right], \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown \end{array}\right]\right\}\) used in this span?
The set is linearly dependent.
The set is linearly independent.
The set spans all of \(\IR^4\text{.}\)
The set fails to span the solution space.
Fact2.7.4.
The coefficients of the free variables in the solution space of a linear system always yield linearly independent vectors that span the solution space.
Which of these is the best choice of basis for this solution space?
\(\displaystyle \{\}\)
\(\displaystyle \{\vec 0\}\)
The basis does not exist
Activity2.7.8.
To create a computer-animated film, an animator first models a scene as a subset of \(\mathbb R^3\text{.}\) Then to transform this three-dimensional visual data for display on a two-dimensional movie screen or television set, the computer could apply a linear tranformation that maps visual information at the point \((x,y,z)\in\mathbb R^3\) onto the pixel located at \((x+y,y-z)\in\mathbb R^2\text{.}\)
(a)
What homoegeneous linear system describes the positions \((x,y,z)\) within the original scene that would be aligned with the pixel \((0,0)\) on the screen?
(b)
Solve this system to describe these locations.
Subsection2.7.2Videos
Figure18.Video: Polynomial and matrix calculations
An \(n \times n\) matrix \(M\) is non-singular if the associated homogeneous system with coefficient matrix \(M\) is consistent with one solution. Assume the matrices in the writing explorations in this section are all non-singular.
Prove that the reduced row echelon form of \(M\) is the identity matrix.
Prove that, for any column vector \(\vec{b} = \left[\begin{array}{c}b_1\\b_2\\ \vdots \\b_n \end{array}\right]\text{,}\) the system of equations given by \(\left[\begin{array}{c|c}M & \vec{b}\end{array}
\right]\) has a unique solution.
Prove that the columns of \(M\) form a basis for \(\mathbb{R}^n\text{.}\)