Write the column matrix b as a linear combination of

angepepagiortzb

angepepagiortzb

Answered question

2021-11-14

Write the column matrix b as a linear combination of the columns of A.
A=[353448],b=[22432]

Answer & Explanation

Ancessitere

Ancessitere

Beginner2021-11-15Added 17 answers

Find 
x=[x1x2] 
such that Ax=b. Then b can be shown as:
b=a1x1+a2x2, where a1,a2 represent the A matrix's columns.
x1[334]+x2[548]=[22432] 
3x1+5x2=22 
3x1+4x2=4 
4x18x2=32 
4x18x2=32 
3x1+4x23x1+5x2=422 
9x2=18 
x2=2 
Find x1
3x1+4x2=4 
3x18=4 
3x1=4+8=12 
x1=4

karton

karton

Expert2023-05-26Added 613 answers

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=[353448]
𝐛=[22432]
To find the scalars x and y, we can set up the equation Ax=𝐛. This can be written as:
[353448][xy]=[22432]
Multiplying the matrices, we get:
[3x+5y3x+4y4x8y]=[22432]
Equating the corresponding entries, we have the following system of equations:
3x+5y=22
3x+4y=4
4x8y=32
To 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

alenahelenash

Expert2023-05-26Added 556 answers

Result:
b=[458/1333t/13278/136t/13992/13+24t/13]
Solution:
Let's represent the columns of matrix A as A1 and A2, and the column matrix b as b. Using the given values:
A=[353448]
b=[22432]
We want to find coefficients x and y such that:
b=xA1+yA2
To 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=44x8y=32
To solve this system, we can use various methods such as substitution, elimination, or matrix operations. Let's solve it using the matrix method.
First, we can represent the system as an augmented matrix:
[35223444832]
Using row operations, we can simplify the augmented matrix to its row-echelon form:
[12801340000]
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's express the solution in terms of the free variable t:
x=8+2t
y=40/133t/13
Substituting these values into the equation b=xA1+yA2, we get:
b=(8+2t)[334]+(40/133t/13)[548]
Simplifying this expression, we obtain the column matrix b as a linear combination of the columns of matrix A:
b=[246t24+6t32][200/13+15t/13160/13+12t/13320/13+24t/13]
Combining like terms, we have:
b=[246t(200/13+15t/13)24+6t(160/13+12t/13)32(320/13+24t/13)]
Simplifying further, we get:
b=[458/1333<br>t/13278/136t/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=[458/1333t/13278/136t/13992/13+24t/13]
user_27qwe

user_27qwe

Skilled2023-05-26Added 375 answers

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+c2𝐚2
where 𝐚1 and 𝐚2 are the columns of matrix A, and c1 and c2 are the coefficients.
Given:
A=[353448]
b=[22432]
We want to find c1 and c2 such that:
[22432]=c1[334]+c2[548]
To solve for c1 and c2, we can set up a system of equations:
22=3c1+5c2
4=3c1+4c2
32=4c18c2
To solve this system, we can represent it in matrix form:
[353448][c1c2]=[22432]
We can solve this system using various methods such as Gaussian elimination, matrix inversion, or least squares. However, for this example, let's solve it using Gaussian elimination:
[35223444832]
35220344048320
35220091800201000
35220091800000
From 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's write the solution in terms of the free variable c2=t:
c1=2t3
c2=t
Thus, the solution for b as a linear combination of the columns of A is:
b=2t3[334]+t[548] where t can take any real value.

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