Use a software program or a graphing utility with matrix capabilities to write [tex]$v$[/tex] as a linear combination of [tex]$u_1, u_2, u_3, u_4$[/tex], and [tex]$u_5$[/tex]. Then verify your solution. (Enter your answer in terms of [tex]$u_1, u_2, u_3, u_4$[/tex], and [tex]$u_5$[/tex].)
[tex]$\begin{array}{l}
v =(5,3,-11,11,9) \
u _1=(1,2,-3,4,-1) \
u _2=(1,2,0,2,1) \
u _3=(0,1,1,1,-4) \
u _4=(2,1,-1,2,1) \
u _5=(0,2,2,-1,-1)
\end{array}$[/tex]
[tex]$v=$[/tex]
Answer (1)
Set up the linear combination equation: v = c 1 u 1 + c 2 u 2 + c 3 u 3 + c 4 u 4 + c 5 u 5 .
Solve the system of linear equations to find the coefficients: c 1 = 2 , c 2 = 1 , c 3 = − 2 , c 4 = 1 , c 5 = − 1 .
Express v as a linear combination of u i using the coefficients found.
The final answer is: v = 2 u 1 + u 2 − 2 u 3 + u 4 − u 5 .
Explanation
Problem Setup We are given the vector v = ( 5 , 3 , − 11 , 11 , 9 ) and the vectors u 1 = ( 1 , 2 , − 3 , 4 , − 1 ) , u 2 = ( 1 , 2 , 0 , 2 , 1 ) , u 3 = ( 0 , 1 , 1 , 1 , − 4 ) , u 4 = ( 2 , 1 , − 1 , 2 , 1 ) , and u 5 = ( 0 , 2 , 2 , − 1 , − 1 ) . Our goal is to express v as a linear combination of u 1 , u 2 , u 3 , u 4 , and u 5 , which means finding coefficients c 1 , c 2 , c 3 , c 4 , c 5 such that v = c 1 u 1 + c 2 u 2 + c 3 u 3 + c 4 u 4 + c 5 u 5 .
Solving for Coefficients This problem can be set up as a system of linear equations. We need to solve for the coefficients c 1 , c 2 , c 3 , c 4 , c 5 . After solving the system of equations, we find the coefficients to be: c 1 = 2 c 2 = 1 c 3 = − 2 c 4 = 1 c 5 = − 1
Expressing v as a Linear Combination Now we can write v as a linear combination of u 1 , u 2 , u 3 , u 4 , and u 5 :
v = 2 u 1 + 1 u 2 − 2 u 3 + 1 u 4 − 1 u 5
Final Answer Therefore, the linear combination is: v = 2 u 1 + u 2 − 2 u 3 + u 4 − u 5
Examples
Linear combinations are used extensively in computer graphics to perform transformations on objects. For example, rotating, scaling, or translating an object can be achieved by expressing the new coordinates of the object's vertices as linear combinations of the original coordinates. This allows for efficient manipulation and animation of 3D models in video games and simulations.
