Write the column matrix b as a linear combination of the columns of A. A=[−35344−8],b=[−22432]

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Write the column matrix b as a linear combination of the columns of A.  A=[−35344−8],b=[−22432]

Ancessitere · Accepted Answer

Find&amp;nbsp;x=[x1x2]&amp;nbsp;such that&amp;nbsp;Ax=b. Then b can be shown as:b=a1x1+a2x2, where&amp;nbsp;a1,a2&amp;nbsp;represent the A matrix&#039;s columns.x1⋅[−334]+x2⋅[54−8]=[−22432]&amp;nbsp;3x1+5x2=−22&amp;nbsp;3x1+4x2=4&amp;nbsp;4x1−8x2=32&amp;nbsp;4x1−8x2=32&amp;nbsp;3x1+4x2−3x1+5x2=4−22&amp;nbsp;9x2=−18&amp;nbsp;x2=−2&amp;nbsp;Find&amp;nbsp;x1:&amp;nbsp;3x1+4x2=4&amp;nbsp;3x1−8=4&amp;nbsp;3x1=4+8=12&amp;nbsp;x1=4

karton · Accepted Answer

We need to find scalars x and y such that the column matrix 𝐛 can be written as a linear combination of the columns of matrix 𝐀.𝐀=[−35344−8]𝐛=[−22432]To find the scalars x and y, we can set up the equation Ax=𝐛. This can be written as:[−35344−8][xy]=[−22432]Multiplying the matrices, we get:[−3x+5y3x+4y4x−8y]=[−22432]Equating the corresponding entries, we have the following system of equations:−3x+5y=−223x+4y=44x−8y=32To solve this system of equations, we can use various methods such as substitution or elimination. The solution to this system of equations gives us the values of x and y. Once we have the values of x and y, we can express 𝐛 as a linear combination of the columns of 𝐀.

alenahelenash · Accepted Answer

Result:b=[−458/13−33t/13278/13−6t/13992/13+24t/13]Solution:Let&#039;s represent the columns of matrix A as A1 and A2, and the column matrix b as b. Using the given values:A=[−35344−8]b=[−22432]We want to find coefficients x and y such that:b=xA1+yA2To solve for x and y, we can set up a system of equations using the columns of matrix A:xA1+yA2=[x(−3)+y(5)x(3)+y(4)x(4)+y(−8)]We can rewrite this system of equations as:{−3x+5y=−223x+4y=44x−8y=32To solve this system, we can use various methods such as substitution, elimination, or matrix operations. Let&#039;s solve it using the matrix method.First, we can represent the system as an augmented matrix:[−35−223444−832]Using row operations, we can simplify the augmented matrix to its row-echelon form:[1−28013−40000]From this row-echelon form, we can see that the last row represents the equation 0 = 0. This means that the system has infinitely many solutions.Let&#039;s express the solution in terms of the free variable t:x=8+2ty=−40/13−3t/13Substituting these values into the equation b=xA1+yA2, we get:b=(8+2t)[−334]+(−40/13−3t/13)[54−8]Simplifying this expression, we obtain the column matrix b as a linear combination of the columns of matrix A:b=[−24−6t24+6t32]−[200/13+15t/13160/13+12t/13−320/13+24t/13]Combining like terms, we have:b=[−24−6t−(200/13+15t/13)24+6t−(160/13+12t/13)32−(−320/13+24t/13)]Simplifying further, we get:b=[−458/13−33&amp;lt;br&amp;gt;t/13278/13−6t/13992/13+24t/13]Hence, the column matrix b can be expressed as a linear combination of the columns of matrix A in the form:b=[−458/13−33t/13278/13−6t/13992/13+24t/13]

user_27qwe · Accepted Answer

To write the column matrix b as a linear combination of the columns of matrix A, we need to find coefficients such that:b=c1𝐚1+c2𝐚2where 𝐚1 and 𝐚2 are the columns of matrix A, and c1 and c2 are the coefficients.Given:A=[−35344−8]b=[−22432]We want to find c1 and c2 such that:[−22432]=c1[−334]+c2[54−8]To solve for c1 and c2, we can set up a system of equations:−22=−3c1+5c24=3c1+4c232=4c1−8c2To solve this system, we can represent it in matrix form:[−35344−8][c1c2]=[−22432]We can solve this system using various methods such as Gaussian elimination, matrix inversion, or least squares. However, for this example, let&#039;s solve it using Gaussian elimination:[−35−223444−832]−35−22034404−8320−35−22009−1800−201000−35−22009−1800000From the row echelon form, we can see that there is a row of zeros. This implies that the system is underdetermined, meaning there are infinitely many solutions. Let&#039;s write the solution in terms of the free variable c2=t:c1=2t3c2=tThus, the solution for b as a linear combination of the columns of A is:b=2t3[−334]+t[54−8] where t can take any real value.

Write the column matrix b as a linear combination of

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